Dynamics of solitons, multi-lumps, and their interactions of the Konopelchenko-Dubrovsky-Kaup-Kupershmidt and Bogoyavlensky-Konopelchenko model in (3+1)-dimensions using the modern advanced approach

Sachin Kumar; Jaionto Karmokar

doi:10.2478/ijmce-2026-0017

Introduction

Nonlinear evolution equations (NLEEs) are commonly used in numerous fields, including soliton theory, nonlinear dynamics, plasma physics, condensed matter physics, optical physics, and fluid dynamics, because most natural systems exhibit nonlinear behavior. In the context of fluid dynamics, these equations describe the evolution of water waves under different heights and pressures. In plasma physics and quantum field theory, NLEEs account for the interactions between particle and wave dispersion, particularly when quantum molecules do not behave linearly across all scales [1–4]. In nonlinear sciences, soliton theory is the most significant and efficient way for finding lump solutions, also known as analytical rational function solutions [5]. Nonlinear partial differential equations involve numerous boundary conditions, expansion, and dispersion terms; these terms often interact with each other, making it difficult to understand the results of the problem. According to Yu and et al. [6], the most significant component of a single wave is the lump wave, which has several characteristics. Manakov and et al. [7] introduced the most basic lump wave solution to the KP equation for two-dimensional soliton solutions in 1977. However, various scientific disciplines have studied lump wave solutions, including oceanography [8], shallow water waves [9], atmospheric [10], nonlinear optics [11], and many more. To generate lump solutions, several efficient techniques have been proposed, including the parameter limit method [12], the bilinear neural network method [13], the Hirota bilinear method [14,15], the Painlevé analysis [16], the Darboux transformation [17,18], and many more. These techniques can help to understand and evaluate different solutions to nonlinear partial differential equations (NPDEs).

Recently, Yang [14] investigated the resonant interaction between 2-solitons and 3-solitons and obtained X-, Y-, and double-X-shaped solitons. Feng and Zhao [19] searched for a suitable expression for the lump solution over the entire plane for a generalized (3+1)-dimensional NPDE. Chen and et al. [20] achieved the lump and breather solutions, as well as their interactions, for the generalized CH-KP equation in hydrodynamics and ocean physics. Ma and et al. [21] employed the Hirota bilinear technique to find the lump solutions of (2+1)-dimensional combined nonlinear equations. Wang and et al. [22] used the symbolic computations to determine solutions to nonlinear wave equations for fluids containing gas bubbles, including lump waves, rogue wave-solitary waves, and mixed lump-solitary waves. Moreover, Yang and Ma [23] applied the Hirota bilinear technique to generate lump solutions for the (2+1)-dimensional BKP equation. He and et al. [24] have investigated the (2+1)-dimensional BLMP and Ito equations for the interaction between lumps and multi-solitons using direct techniques. Ma and Zhou [5] introduced the quadratic function method to generate lump solutions for NPDEs. Using the quadratic function method, Zhou and et al. [25] obtained lump solutions to the (3+1)-dimensional generalized CBS equation. Inspired by Hirota’s direct technique, we focus on lump soliton solutions and their collisions by combining quadratic functions with trigonometric and exponential functions.

In this study, we investigate a (3+1)-dimensional generalized NPDEs, also known as the KDKK and BK equations introduced by Feng and Zhao [19], as follows: 1 $κ_{1} \frac{\partial u}{\partial t} + κ_{2} (\frac{\partial^{3} u}{\partial x^{2} \partial z} + 3 u \frac{\partial u}{\partial z} + 3 \frac{\partial u}{\partial x} \partial^{- 1} u_{z}) + κ_{3} (\frac{\partial^{3} u}{\partial x^{2} \partial y} + 3 u \frac{\partial u}{\partial y} + 3 \frac{\partial u}{\partial x} \partial^{- 1} u_{y}) + κ_{4} \frac{\partial u}{\partial x} + κ_{5} \frac{\partial u}{\partial y} + κ_{6} \frac{\partial u}{\partial z} = 0,$ {\kappa _1}{{\partial u} \over {\partial t}} + {\kappa _2}\left( {{{{\partial ^3}u} \over {\partial {x^2}\partial z}} + 3u{{\partial u} \over {\partial z}} + 3{{\partial u} \over {\partial x}}{\partial ^{ - 1}}{u_z}} \right) + {\kappa _3}\left( {{{{\partial ^3}u} \over {\partial {x^2}\partial y}} + 3u{{\partial u} \over {\partial y}} + 3{{\partial u} \over {\partial x}}{\partial ^{ - 1}}{u_y}} \right) + {\kappa _4}{{\partial u} \over {\partial x}} + {\kappa _5}{{\partial u} \over {\partial y}} + {\kappa _6}{{\partial u} \over {\partial z}} = 0, where $\partial^{- 1} u_{z} = \int \frac{\partial u}{\partial z} d x, \partial^{- 1} u_{y} = \int \frac{\partial u}{\partial y} d x$ {\partial ^{ - 1}}{u_z} = \int {{{\partial u} \over {\partial z}}} dx,{\partial ^{ - 1}}{u_y} = \int {{{\partial u} \over {\partial y}}} dx, and κ_i(1 ≤ i ≤ 6) are nonzero real constants. The presence of constant and free parameters is crucial for determining the exact solution of equation (1). Table 1 presents the generalized well-known equations derived from equation (1) by variation of κ_i.

Table 1

Particular cases of equation (1) found in ocean dynamics and related fields.

Variation of κ_i	PDEs	Special form
κ₁ = κ₂ = 1, κ₃ = κ₄ = κ₅ = κ₆ = 0, z = x	u_t + u_xxx + 6uu_x = 0	KdV equation [26]
κ₁ = 1, κ₂ = a₁, κ₃ = a₅, κ₅ = 0, κ₄ + κ₆ = 0, z = x	u_t + a₁ u_xxx + $\frac{1}{2} a_{2} {(u^{2})}_{x}$ {1 \over 2}{a_2}{\left( {{u^2}} \right)_x} + a₃u_xxxxx + a₄v_y + a₅u_xxy + a₆(u_xv + uv_x) + a₇(uu_xx)_x + $\frac{1}{3} a_{8} {(u^{3})}_{x} = 0$ {1 \over 3}{a_8}{\left( {{u^3}} \right)_x} = 0, v_x = u_y, for a₂ = 6a₁, a₆ = 3a₅, a₃ = a₄ = a₇ = a₈ = 0	(2+1)-D gKDKK equation [27]
κ₁ = 1, κ₂ = b₁, κ₃ = b₂, κ₄ = b₃, κ₅ = b₄, κ₆ = 0, b₅ = 0, z = x	u_t + $6 b_{1} (u u_{x} + \frac{1}{6} u_{x x x})$ 6{b_1}\left( {u{u_x} + {1 \over 6}{u_{xxx}}} \right) + $3 b_{2} (\frac{1}{3} u_{x x y} + u u_{y} + u_{x} v_{y})$ 3{b_2}\left( {{1 \over 3}{u_{xxy}} + u{u_y} + {u_x}{v_y}} \right) + b₃u_x + b₄u_y + b₅V_yy = 0, V_x = u	(2+1)-D gBK equation [28]

The purpose of this research is to present novel insights into the collision of waves, such as lump, periodic, solitary, and breather waves in a generalized KDKK and BK equations through the Hirota bilinear technique. The motivation behind this technique has several advantages. First, this method is a very efficient, accurate, reliable, and sophisticated way to determine the exact solution of the NPDEs. Second, this method does not require inverse scattering integration, which makes it easier to depict soliton solutions. Third, this method is the easiest to handle algebraically because it transforms the governing NPDEs into a bilinear form. Finally, once the bilinear form is known, it provides a systematic technique that can often be used to solve for N-soliton solutions.

In addition, we introduce the analysis of bifurcation and equilibrium states by considering a travelling wave transformation of the governing equation to obtain a 2D planar dynamical structure. The solution to the system is related to the Hamiltonian function, which is shown in the figure.

The Hamiltonian system is constructed more specifically using Galilean transformations, which emphasize the system’s periodic and chaotic behavior [29–32]. For a given parameter, a dynamical system can exhibit the phenomenon of multistability, which consists of multiple distinct dynamical behaviors under different initial conditions [32, 33]. Sensitivity analysis makes it easier to understand how a dynamic system behaves under different initial conditions.

To the best of our knowledge, equation (1) has not been studied in the context of collisions between periodic and lump waves, collisions between single-lump and solitary waves, or collisions between lump and breather waves. Furthermore, this equation has never been investigated through bifurcation analysis, equilibrium conditions, and sensitivity analysis. This article fills this gap by using a bifurcation-based approach to analyze soliton interaction, water wave dynamics, the effect of wind on equilibrium structures, stability, and soliton dynamics. It has special significance due to the enhanced physical formulation of the KDKK and BK equations, as well as the methodological innovation of applying bifurcation and sensitivity analysis. This combination provides a new understanding of how external attraction and dissipation affect nonlinear wave dynamics.

This paper is outlined as follows: In Section 2, the Painlevé test is employed to examine the integrability of equation (1). Section 3 discusses the proposed method. Section 4 explores the wave-to-wave collision structures and their solutions using different ansatz functions. Section 5 presents the physical interpretation of the findings. Section 6 discusses the equilibrium state and bifurcation analysis of a dynamic system, which shows the phase portrait and Hamiltonian function graphics. Section 7 covers the sensitivity analysis of a dynamic system for different initial conditions. Finally, the findings are summarized and presented as a conclusion in Section 8.

Painlevé analysis

This section provides an outline of the Painlevé test, which is an effective technique for checking whether a nonlinear PDE can be solved in an integrable manner [34, 35]. If a nonlinear system passes the Painlevé test, that is, if each movable singular point of the solutions is a simple pole, then it is deemed integrable. Weiss and et al. [36] developed this technique to check the integrability of nonlinear PDEs by examining the integrability criteria. This process is described in three steps: first, the behavior of the leading-order is examined; second, the resonances are identified; and third, the conditions at the resonance points are verified.

We assume that the field u in equation (1) is expanded in a Laurent series about the singular manifold h(x, y, z, t) = 0, to perform the integrability test as 2 $u (x, y, z, t) = \sum_{q = 0}^{\infty} u_{q} h^{q + Ω},$ u(x,y,z,t) = \sum\limits_{q = 0}^\infty {{u_q}} {h^{q + \Omega }}, where Ω is an integer and u_q = u_q(x, y, z, t); q = 0, 1, 2, ⋯ are arbitrary functions, respectively. Substituting equation (2) into equation (1) through a leading-order analysis based on the dominant terms, we get $Ω = - 2, u_{0} = - 2 h_{x}^{2} .$ \Omega = - 2,\quad {u_0} = - 2h_x^2.

Next, to obtain resonance, we take $u = u_{0} h^{- 2} + u_{q} h^{q - 2} .$ u = {u_0}{h^{ - 2}} + {u_q}{h^{q - 2}}.

Substituting these into equation (1) and collecting the coefficients of the dominant terms gives the characteristic relationship for resonance, which is: 3 $((q - 2) (q - 3) (q - 4) - 12 (q - 4)) h_{x}^{2} (κ_{2} h_{z} + κ_{3} h_{y}) = 0.$ ((q - 2)(q - 3)(q - 4) - 12(q - 4))h_x^2\left( {{\kappa _2}{h_z} + {\kappa _3}{h_y}} \right) = 0.

Since h(x, y, z, t) and its derivatives are not equal to zero, the resonances of equation (1) are $q = - 1, 4, 6.$ q = - 1,\quad 4,\quad 6.

The resonance q = –1 indicates the random selection of the singular manifold h(x, y, z, t) = 0. The explicit expression for u_q indicates the existence of sufficient arbitrary functions for q = 4 and q = 6 resonances. The condition of compatibility is fulfilled by positive resonance. Therefore, the tested KDKK and BK equations (1) pass the Painlevé test for integrability.

Hirota’s bilinear form of the governing equation

Suppose Ψ_i is the phase of equation (1), such that 4 $Ψ_{i} = α_{i} x + β_{i} y + γ_{i} z - δ_{i} t + ϑ_{i},$ {\Psi _i} = {\alpha _i}x + {\beta _i}y + {\gamma _i}z - {\delta _i}t + {\vartheta _i}, where α_i, β_i, γ_i, and ϑ_i are real parameters. By substituting u = e^Ψ_i into equation (1) and considering the linear terms, we determine that the dispersion δ_i is 5 $δ_{i} = \frac{β_{i} α_{i}^{2} κ_{3} + α_{i}^{2} γ_{i} κ_{2} + β_{i} κ_{5} + α_{i} κ_{4} + γ_{i} κ_{6}}{κ_{1}} .$ {\delta _i} = {{{\beta _i}\alpha _i^2{\kappa _3} + \alpha _i^2{\gamma _i}{\kappa _2} + {\beta _i}{\kappa _5} + {\alpha _i}{\kappa _4} + {\gamma _i}{\kappa _6}} \over {{\kappa _1}}}.

Now, consider a logarithmic transformation (also known as the Cole-Hopf transformation) for the dependent variable u = u(x, y, z, t) as 6 $u = \frac{\partial^{2}}{\partial x^{2}} (2 \ln f) .$ u = {{{\partial ^2}} \over {\partial {x^2}}}(2\ln f).

Let f = f(x, y, z, t) be an auxiliary function, and rewrite equation (6) in another form as follows: 7 $u = V_{x x}, V = 2 \ln f .$ u = {V_{xx}},\quad V = 2\ln f.

Substituting equation (7) into equation (1) and integrating with respect to x, we get 8 $κ_{1} V_{x t} + κ_{2} (3 V_{x x} V_{x z} + V_{x x x z}) + κ_{3} (3 V_{x x} V_{x y} + V_{x x x y}) + κ_{4} V_{x x} + κ_{5} V_{x y} + κ_{6} V_{x z} = 0.$ {\kappa _1}{V_{xt}} + {\kappa _2}\left( {3{V_{xx}}{V_{xz}} + {V_{xxxz}}} \right) + {\kappa _3}\left( {3{V_{xx}}{V_{xy}} + {V_{xxxy}}} \right) + {\kappa _4}{V_{xx}} + {\kappa _5}{V_{xy}} + {\kappa _6}{V_{xz}} = 0.

Substituting the derivatives of the second term of equation (7) into equation (8), we get 9 $\begin{array}{r} κ_{1} (2 f f_{t x} - 2 f_{t} f_{x}) + κ_{2} (2 f f_{x x x z} - 6 f_{x} f_{x x z} + 6 f_{x x} f_{x z} - 2 f_{x x x} f_{z}) + \\ κ_{3} (2 f f_{x x x y} - 6 f_{x} f_{x x y} + 6 f_{x x} f_{x y} - 2 f_{x x x} f_{y}) + κ_{4} (2 f f_{x x} - 2 f_{x}^{2}) + \\ κ_{5} (2 f f_{x y} - 2 f_{x} f_{y}) + κ_{6} (2 f f_{x z} - 2 f_{x} f_{z}) = 0. \end{array}$ \matrix{ \hfill {{\kappa _1}\left( {2f{f_{tx}} - 2{f_t}{f_x}} \right) + {\kappa _2}\left( {2f{f_{xxxz}} - 6{f_x}{f_{xxz}} + 6{f_{xx}}{f_{xz}} - 2{f_{xxx}}{f_z}} \right) + } \cr \hfill {{\kappa _3}\left( {2f{f_{xxxy}} - 6{f_x}{f_{xxy}} + 6{f_{xx}}{f_{xy}} - 2{f_{xxx}}{f_y}} \right) + {\kappa _4}\left( {2f{f_{xx}} - 2f_x^2} \right) + } \cr \hfill {{\kappa _5}\left( {2f{f_{xy}} - 2{f_x}{f_y}} \right) + {\kappa _6}\left( {2f{f_{xz}} - 2{f_x}{f_z}} \right) = 0.} \cr }

This equation represents a bilinear equation that is transformed into Hirota bilinear form using a bilinear differential operator. Hirota [15] designed the bilinear differential operator D_r, where r = x, y, z, t. It can be expressed as $\begin{array}{l} D_{x}^{p_{1}} D_{y}^{p_{2}} D_{z}^{p_{3}} D_{t}^{p_{4}} M \cdot N & = & {(\frac{\partial}{\partial x} - \frac{\partial}{\partial x^{'}})}^{p_{1}} {(\frac{\partial}{\partial y} - \frac{\partial}{\partial y^{'}})}^{p_{2}} {(\frac{\partial}{\partial z} - \frac{\partial}{\partial z^{'}})}^{p_{3}} {(\frac{\partial}{\partial t} - \frac{\partial}{\partial t^{'}})}^{p_{4}} \\ {M (x, y, z, t) N (x^{'}, y^{'}, z^{'}, t^{'}) |}_{x^{'} = x, y^{'} = y, z^{'} = z, t^{'} = t}, \end{array}$ \matrix{ {D_x^{{p_1}}D_y^{{p_2}}D_z^{{p_3}}D_t^{{p_4}}M\cdotN} \hfill & = \hfill & {{{\left( {{\partial \over {\partial x}} - {\partial \over {\partial {x^\prime }}}} \right)}^{{p_1}}}{{\left( {{\partial \over {\partial y}} - {\partial \over {\partial {y^\prime }}}} \right)}^{{p_2}}}{{\left( {{\partial \over {\partial z}} - {\partial \over {\partial {z^\prime }}}} \right)}^{{p_3}}}{{\left( {{\partial \over {\partial t}} - {\partial \over {\partial {t^\prime }}}} \right)}^{{p_4}}}} \hfill \cr {} \hfill & {} \hfill & {{{\left. {M(x,y,z,t)N\left( {{x^\prime },{y^\prime },{z^\prime },{t^\prime }} \right)} \right|}_{{x^\prime } = x,{y^\prime } = y,{z^\prime } = z,{t^\prime } = t'}}} \hfill \cr } where p₁, p₂, p₃, p₄ are positive integers and x′, y′, z′, t′ are formal variables. Using this definition of the D-operator, we obtain the necessary operators, such as 10 $\begin{array}{l} D_{x} D_{j} f \cdot f & = & 2 (f f_{x j} - f_{x} f_{j}); j = y, z, t, \\ D_{x}^{2} f \cdot f & = & 2 (f f_{x x} - f_{x}^{2}), \\ D_{x}^{3} D_{j} f \cdot f & = & 2 (f f_{x x x j} - 3 f_{x} f_{x x j} + 3 f_{x x} f_{x j} - f_{x x x} f_{j}); j = y, z . \end{array}$ \matrix{ {{D_x}{D_j}f\cdotf} \hfill & = \hfill & {2\left( {f{f_{xj}} - {f_x}{f_j}} \right);j = y,z,t,} \hfill \cr {D_x^2f\cdotf} \hfill & = \hfill & {2\left( {f{f_{xx}} - f_x^2} \right),} \hfill \cr {D_x^3{D_j}f\cdotf} \hfill & = \hfill & {2\left( {f{f_{xxxj}} - 3{f_x}{f_{xxj}} + 3{f_{xx}}{f_{xj}} - {f_{xxx}}{f_j}} \right);j = y,z.} \hfill \cr }

Therefore, the bilinear form of equation (9) is 11 $[κ_{1} D_{x} D_{t} + κ_{2} D_{x}^{3} D_{z} + κ_{3} D_{x}^{3} D_{y} + κ_{4} D_{x}^{2} + κ_{5} D_{x} D_{y} + κ_{6} D_{x} D_{z}] f \cdot f = 0.$ \left[ {{\kappa _1}{D_x}{D_t} + {\kappa _2}D_x^3{D_z} + {\kappa _3}D_x^3{D_y} + {\kappa _4}D_x^2 + {\kappa _5}{D_x}{D_y} + {\kappa _6}{D_x}{D_z}} \right]f\cdotf = 0.

Therefore, if the auxiliary function f satisfies equation (11), then equation (6) will be considered as a solution of KDKK and BK equations (1). In the following section, we investigate an expression for an ansatz function that leads to the general solutions for solitary waves, breather waves, lump waves, periodic waves, and their collisions in detail.

Wave-to-wave collisions with different ansatz functions and their solutions

4.1

Lump wave solutions of KDKK and BK equations

An expression for the auxiliary function f is considered to find the lump wave solution of the KDKK and BK equations (1): 12 $f = {(l_{1} x + m_{1} y + n_{1} z + w_{1} t + δ_{1})}^{2} + {(l_{2} x + m_{2} y + n_{2} z + w_{2} t + δ_{2})}^{2} + η,$ f = {\left( {{l_1}x + {m_1}y + {n_1}z + {w_1}t + {\delta _1}} \right)^2} + {\left( {{l_2}x + {m_2}y + {n_2}z + {w_2}t + {\delta _2}} \right)^2} + \eta , where l₁, l₂, m₁, m₂, n₁, n₂, w₁, w₂, δ₁, δ₂, and η are free parameters to be determined later. Substituting equation (12) into equation (11) and equating all the coefficients of x, y, z, t, and their multiples equal to zero. Then, a system of algebraic equations consisting of l₁, l₂, m₁, m₂, n₁, n₂, w₁, w₂, δ₁, δ₂, and η is obtained. By solving the system of equations using symbolic software Maple, we obtain the values of some parameters:

Case I

l_{2} = 0, m_{1} = - \frac{κ_{2} (w_{1} κ_{1} + l_{1} κ_{4})}{κ_{2} κ_{5} - κ_{3} κ_{6}}, n_{1} = \frac{κ_{3} (w_{1} κ_{1} + l_{1} κ_{4})}{κ_{2} κ_{5} - κ_{3} κ_{6}}, w_{2} = - \frac{m_{2} κ_{5} + n_{2} κ_{6}}{κ_{1}} .

{l_2} = 0,{m_1} = - {{{\kappa _2}\left( {{w_1}{\kappa _1} + {l_1}{\kappa _4}} \right)} \over {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}}},{n_1} = {{{\kappa _3}\left( {{w_1}{\kappa _1} + {l_1}{\kappa _4}} \right)} \over {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}}},{w_2} = - {{{m_2}{\kappa _5} + {n_2}{\kappa _6}} \over {{\kappa _1}}}.

Substituting the parameter values of equation (13) into equation (12), we attain 14 $\begin{array}{l} f & = & {(l_{1} x - \frac{κ_{2} (l_{1} κ_{4} + w_{1} κ_{1}) y}{κ_{2} κ_{5} - κ_{3} κ_{6}} + \frac{κ_{3} (l_{1} κ_{4} + w_{1} κ_{1}) z}{κ_{2} κ_{5} - κ_{3} κ_{6}} + w_{1} t + δ_{1})}^{2} + \\ {(m_{2} y + n_{2} z - \frac{t (m_{2} κ_{5} + n_{2} κ_{6})}{κ_{1}} + δ_{2})}^{2} + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( {{l_1}x - {{{\kappa _2}\left( {{l_1}{\kappa _4} + {w_1}{\kappa _1}} \right)y} \over {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}}} + {{{\kappa _3}\left( {{l_1}{\kappa _4} + {w_1}{\kappa _1}} \right)z} \over {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}}} + {w_1}t + {\delta _1}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {{{\left( {{m_2}y + {n_2}z - {{t\left( {{m_2}{\kappa _5} + {n_2}{\kappa _6}} \right)} \over {{\kappa _1}}} + {\delta _2}} \right)}^2} + \eta .} \hfill \cr }

Substituting the expression (14) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which represent the lump-type wave solution. Choosing the particular value of the free parameters: l₁ = 1, w₁ = 1, m₂ = 1, n₂ = 1, δ₁ = 1, δ₂ = 1, κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –2, κ₅ = –1, κ₆ = –2, and η = 1. Then, the solutions are calculated and simplified. Finally, we obtain the specific lump wave solution of the governing equation (1) is 15 $u = \frac{4 (- x^{2} + 8 t^{2} - 2 x y + 2 x z - 2 x t + 4 y t + 8 z t + 4 y z - 2 x + 4 z + 4 t + 1)}{{(x^{2} + 2 y^{2} + 2 z^{2} + 10 t^{2} + 2 x y - 2 x z + 2 x t + 8 y t + 4 z t + 2 x + 4 y + 8 t + 3)}^{2}} .$ u = {{4\left( { - {x^2} + 8{t^2} - 2xy + 2xz - 2xt + 4yt + 8zt + 4yz - 2x + 4z + 4t + 1} \right)} \over {{{\left( {{x^2} + 2{y^2} + 2{z^2} + 10{t^2} + 2xy - 2xz + 2xt + 8yt + 4zt + 2x + 4y + 8t + 3} \right)}^2}}}.

The wave profiles of the lump wave solutions of equation (15) are shown in Fig. 1 through contour plots, 2D, and 3D profiles.

Case II

\begin{array}{l} m_{1} & = & - \frac{l_{1} κ_{4} + n_{1} κ_{6} + w_{1} κ_{1}}{κ_{5}}, \\ m_{2} & = & - \frac{κ_{3} κ_{4} κ_{6} l_{1}^{2} - κ_{2} κ_{5} κ_{6} l_{1} n_{1} + κ_{3} κ_{6}^{2} l_{1} n_{1} + κ_{1} κ_{3} κ_{6} l_{1} w_{1} + κ_{2} κ_{4} κ_{5} l_{2}^{2} + κ_{1} κ_{2} κ_{5} l_{2} w_{2}}{κ_{5} l_{2} (κ_{2} κ_{5} - κ_{3} κ_{6})}, \\ n_{2} & = & \frac{κ_{3} κ_{4} l_{1}^{2} - κ_{2} κ_{5} l_{1} n_{1} + κ_{3} κ_{6} n_{1} l_{1} + κ_{1} κ_{3} l_{1} w_{1} + κ_{3} κ_{4} l_{2}^{2} + κ_{1} κ_{3} w_{2} l_{2}}{(κ_{2} κ_{5} - κ_{3} κ_{6}) l_{2}} . \end{array}

\matrix{ {{m_1}} \hfill & = \hfill & { - {{{l_1}{\kappa _4} + {n_1}{\kappa _6} + {w_1}{\kappa _1}} \over {{\kappa _5}}},} \hfill \cr {{m_2}} \hfill & = \hfill & { - {{{\kappa _3}{\kappa _4}{\kappa _6}l_1^2 - {\kappa _2}{\kappa _5}{\kappa _6}{l_1}{n_1} + {\kappa _3}\kappa _6^2{l_1}{n_1} + {\kappa _1}{\kappa _3}{\kappa _6}{l_1}{w_1} + {\kappa _2}{\kappa _4}{\kappa _5}l_2^2 + {\kappa _1}{\kappa _2}{\kappa _5}{l_2}{w_2}} \over {{\kappa _5}{l_2}\left( {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}} \right)}},} \hfill \cr {{n_2}} \hfill & = \hfill & {{{{\kappa _3}{\kappa _4}l_1^2 - {\kappa _2}{\kappa _5}{l_1}{n_1} + {\kappa _3}{\kappa _6}{n_1}{l_1} + {\kappa _1}{\kappa _3}{l_1}{w_1} + {\kappa _3}{\kappa _4}l_2^2 + {\kappa _1}{\kappa _3}{w_2}{l_2}} \over {\left( {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}} \right){l_2}}}.} \hfill \cr }

Substituting the parameter values of equation (16) into equation (12), we attain 17 $\begin{array}{l} f & = & {(l_{1} x - \frac{(l_{1} κ_{4} + n_{1} κ_{6} + w_{1} κ_{1}) y}{κ_{5}} + n_{1} z + w_{1} t + δ_{1})}^{2} + \\ (l_{2} x - \frac{(κ_{3} κ_{4} κ_{6} l_{1}^{2} - κ_{2} κ_{5} κ_{6} l_{1} n_{1} + κ_{3} κ_{6}^{2} l_{1} n_{1} + κ_{1} κ_{3} κ_{6} l_{1} w_{1} + κ_{2} κ_{4} κ_{5} l_{2}^{2} + κ_{1} κ_{2} κ_{5} l_{2} w_{2}) y}{κ_{5} l_{2} (κ_{2} κ_{5} - κ_{3} κ_{6})} + \\ {\frac{(κ_{3} κ_{4} l_{1}^{2} - κ_{2} κ_{5} l_{1} n_{1} + κ_{3} κ_{6} n_{1} l_{1} + κ_{1} κ_{3} l_{1} w_{1} + κ_{3} κ_{4} l_{2}^{2} + κ_{1} κ_{3} w_{2} l_{2}) z}{(κ_{2} κ_{5} - κ_{3} κ_{6}) l_{2}} + w_{2} t + δ_{2})}^{2} + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( {{l_1}x - {{\left( {{l_1}{\kappa _4} + {n_1}{\kappa _6} + {w_1}{\kappa _1}} \right)y} \over {{\kappa _5}}} + {n_1}z + {w_1}t + {\delta _1}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {\left( {{l_2}x - {{\left( {{\kappa _3}{\kappa _4}{\kappa _6}l_1^2 - {\kappa _2}{\kappa _5}{\kappa _6}{l_1}{n_1} + {\kappa _3}\kappa _6^2{l_1}{n_1} + {\kappa _1}{\kappa _3}{\kappa _6}{l_1}{w_1} + {\kappa _2}{\kappa _4}{\kappa _5}l_2^2 + {\kappa _1}{\kappa _2}{\kappa _5}{l_2}{w_2}} \right)y} \over {{\kappa _5}{l_2}\left( {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}} \right)}} + } \right.} \hfill \cr {} \hfill & {} \hfill & {{{\left. {{{\left( {{\kappa _3}{\kappa _4}l_1^2 - {\kappa _2}{\kappa _5}{l_1}{n_1} + {\kappa _3}{\kappa _6}{n_1}{l_1} + {\kappa _1}{\kappa _3}{l_1}{w_1} + {\kappa _3}{\kappa _4}l_2^2 + {\kappa _1}{\kappa _3}{w_2}{l_2}} \right)z} \over {\left( {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}} \right){l_2}}} + {w_2}t + {\delta _2}} \right)}^2} + \eta .} \hfill \cr }

Substituting the expression (17) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which represent the lump-type wave solution. Choosing the particular value of the free parameters: l₁ = 1, n₁ = 1, w₁ = 1, l₂ = 1, w₂ = 1, δ₁ = –1, δ₂ = 1, η = 1, κ₁ = 2, κ₂ = 2, κ₃ = 1, κ₄ = –2, κ₅ = –1, and κ₆ = 1. Then, the solutions are calculated and simplified. Finally, we obtain the specific lump wave solution of the governing equation (1) is 18 $u = - \frac{8 (2 x^{2} - 2 y^{2} - 2 z^{2} + 2 t^{2} + 4 x t - 4 y z + 4 y + 4 z - 3)}{{(2 x^{2} + 2 y^{2} + 2 z^{2} + 2 t^{2} + 4 x t + 4 y z - 4 y - 4 z + 3)}^{2}} .$ u = - {{8\left( {2{x^2} - 2{y^2} - 2{z^2} + 2{t^2} + 4xt - 4yz + 4y + 4z - 3} \right)} \over {{{\left( {2{x^2} + 2{y^2} + 2{z^2} + 2{t^2} + 4xt + 4yz - 4y - 4z + 3} \right)}^2}}}.

In Fig. 2, the wave profiles of the lump wave solutions of equation (18) are shown for different values of z through contour plots, 2D, and 3D profiles.

4.2

Collision of waves: single-lump wave and solitary wave

Consider a function that combines quadratic and exponential functions to examine the effect of a collision between a single lump and a solitary wave on the governing equation (1). For example, 19 $f = {(l_{1} x + m_{1} y + n_{1} z + w_{1} t + δ_{1})}^{2} + {(l_{2} x + m_{2} y + n_{2} z + w_{2} t + δ_{2})}^{2} + e x p (l_{3} x + m_{3} y + n_{3} z + w_{3} t) + η,$ f = {\left( {{l_1}x + {m_1}y + {n_1}z + {w_1}t + {\delta _1}} \right)^2} + {\left( {{l_2}x + {m_2}y + {n_2}z + {w_2}t + {\delta _2}} \right)^2} + \exp \left( {{l_3}x + {m_3}y + {n_3}z + {w_3}t} \right) + \eta , where l_r, m_r, n_r, w_r, δ₁, δ₂, and η are free parameters with r = 1,2,3 to be determined. Substituting equation (19) into equation (11) and equating all the coefficients of x, y, z, t, and their multiples equal to zero. Then, a system of algebraic equations consisting of l_r, m_r, n_r, w_r, δ₁, δ₂, and η with 1 ≤ r ≤ 3 is obtained. By solving the system of equations using symbolic software Maple, we obtain the values of some parameters:

Case I

\begin{array}{l} l_{1} = 0, l_{2} = - \frac{- κ_{2} n_{2} κ_{5} + κ_{6} n_{2} κ_{3} + w_{2} κ_{1} κ_{3}}{κ_{3} κ_{4}}, m_{1} = - \frac{κ_{2} n_{1}}{κ_{3}}, m_{2} = - \frac{κ_{2} n_{2}}{κ_{3}}, \\ m_{3} = - \frac{κ_{2} n_{3}}{κ_{3}}, w_{1} = \frac{n_{1} (κ_{2} κ_{5} - κ_{3} κ_{6})}{κ_{3} κ_{1}}, w_{3} = \frac{- l_{3} κ_{4} κ_{3} + κ_{2} κ_{5} n_{3} - κ_{3} n_{3} κ_{6}}{κ_{3} κ_{1}} . \end{array}

\matrix{ {{l_1} = 0,{l_2} = - {{ - {\kappa _2}{n_2}{\kappa _5} + {\kappa _6}{n_2}{\kappa _3} + {w_2}{\kappa _1}{\kappa _3}} \over {{\kappa _3}{\kappa _4}}},{m_1} = - {{{\kappa _2}{n_1}} \over {{\kappa _3}}},{m_2} = - {{{\kappa _2}{n_2}} \over {{\kappa _3}}},} \hfill \cr {{m_3} = - {{{\kappa _2}{n_3}} \over {{\kappa _3}}},{w_1} = {{{n_1}\left( {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}} \right)} \over {{\kappa _3}{\kappa _1}}},{w_3} = {{ - {l_3}{\kappa _4}{\kappa _3} + {\kappa _2}{\kappa _5}{n_3} - {\kappa _3}{n_3}{\kappa _6}} \over {{\kappa _3}{\kappa _1}}}.} \hfill \cr }

Substituting the parameter values of equation (20) into equation (19), we attain 21 $\begin{array}{l} f & = & {(- \frac{κ_{2} n_{1} y}{κ_{3}} + n_{1} z + \frac{t n_{1} (κ_{2} κ_{5} - κ_{3} κ_{6})}{κ_{3} κ_{1}} + δ_{1})}^{2} + \\ {(- \frac{(- κ_{2} n_{2} κ_{5} + κ_{6} n_{2} κ_{3} + w_{2} κ_{1} κ_{3}) x}{κ_{3} κ_{4}} - \frac{κ_{2} n_{2} y}{κ_{3}} + n_{2} z + w_{2} t + δ_{2})}^{2} + \\ e^{l_{3} x - \frac{κ_{2} n_{3} y}{κ_{3}} + n_{3} z + \frac{t (- l_{3} κ_{4} κ_{3} + κ_{2} κ_{5} n_{3} - κ_{3} n_{3} κ_{6})}{κ_{3} κ_{1}}} + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( { - {{{\kappa _2}{n_1}y} \over {{\kappa _3}}} + {n_1}z + {{t{n_1}\left( {{\kappa _2}{\kappa _5} - {\kappa _3}{\kappa _6}} \right)} \over {{\kappa _3}{\kappa _1}}} + {\delta _1}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {{{\left( { - {{\left( { - {\kappa _2}{n_2}{\kappa _5} + {\kappa _6}{n_2}{\kappa _3} + {w_2}{\kappa _1}{\kappa _3}} \right)x} \over {{\kappa _3}{\kappa _4}}} - {{{\kappa _2}{n_2}y} \over {{\kappa _3}}} + {n_2}z + {w_2}t + {\delta _2}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {{e^{{l_3}x - {{{\kappa _2}{n_3}y} \over {{\kappa _3}}} + {n_3}z + {{t\left( { - {l_3}{\kappa _4}{\kappa _3} + {\kappa _2}{\kappa _5}{n_3} - {\kappa _3}{n_3}{\kappa _6}} \right)} \over {{\kappa _3}{\kappa _1}}}}} + \eta .} \hfill \cr }

Substituting the expression (21) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which reveals the collisions of the single-lump and solitary-wave solutions. Taking the specific value of the free parameters: l₃ = 1, n₁ = n₂ = n₃ = 1, w₂ = 1, δ₁ = δ₂ = 1, κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –2, κ₅ = κ₆ = 1, and η = 1. The specific wave solution of the governing equation (1) is 22 $\begin{array}{l} u & = & \frac{1 + 2 e^{ϕ_{1}}}{e^{ϕ_{1}} + t^{2} + (x - 2 y + 2 z + 2) t + \frac{x^{2}}{4} + (1 - y + z) x + 2 y^{2} - 4 (z + 1) y + 2 z^{2} + 4 z + 3} - \\ \frac{\frac{1}{2} {(x - 2 y + 2 z + 2 t + 2 + 2 e^{ϕ_{1}})}^{2}}{{(e^{ϕ_{1}} + t^{2} + (x - 2 y + 2 z + 2) t + \frac{x^{2}}{4} + (1 - y + z) x + 2 y^{2} - 4 (z + 1) y + 2 z^{2} + 4 z + 3)}^{2}}, \end{array}$ \matrix{ u \hfill & = \hfill & {{{1 + 2{e^{{\phi _1}}}} \over {{e^{{\phi _1}}} + {t^2} + (x - 2y + 2z + 2)t + {{{x^2}} \over 4} + (1 - y + z)x + 2{y^2} - 4(z + 1)y + 2{z^2} + 4z + 3}} - } \hfill \cr {} \hfill & {} \hfill & {{{{1 \over 2}{{\left( {x - 2y + 2z + 2t + 2 + 2{e^{{\phi _1}}}} \right)}^2}} \over {{{\left( {{e^{{\phi _1}}} + {t^2} + (x - 2y + 2z + 2)t + {{{x^2}} \over 4} + (1 - y + z)x + 2{y^2} - 4(z + 1)y + 2{z^2} + 4z + 3} \right)}^2}}},} \hfill \cr } where ϕ₁ = x – y + z + 2t. Its propagation velocity along the xy-plane is shown in Fig. 3.

Case II

\begin{array}{l} m_{1} = - \frac{κ_{2} n_{1}}{κ_{3}}, m_{2} = - \frac{κ_{2} n_{2}}{κ_{3}}, m_{3} = - \frac{κ_{2} n_{3}}{κ_{3}}, w_{1} = \frac{κ_{2} κ_{5} n_{1} - κ_{3} l_{1} κ_{4} - κ_{3} n_{1} κ_{6}}{κ_{3} κ_{1}}, \\ w_{2} = \frac{κ_{2} n_{2} κ_{5} - l_{2} κ_{3} κ_{4} - κ_{6} n_{2} κ_{3}}{κ_{3} κ_{1}}, w_{3} = \frac{κ_{2} κ_{5} n_{3} - l_{3} κ_{4} κ_{3} - κ_{3} n_{3} κ_{6}}{κ_{3} κ_{1}} . \end{array}

\matrix{ {{m_1} = - {{{\kappa _2}{n_1}} \over {{\kappa _3}}},{m_2} = - {{{\kappa _2}{n_2}} \over {{\kappa _3}}},{m_3} = - {{{\kappa _2}{n_3}} \over {{\kappa _3}}},{w_1} = {{{\kappa _2}{\kappa _5}{n_1} - {\kappa _3}{l_1}{\kappa _4} - {\kappa _3}{n_1}{\kappa _6}} \over {{\kappa _3}{\kappa _1}}},} \hfill \cr {{w_2} = {{{\kappa _2}{n_2}{\kappa _5} - {l_2}{\kappa _3}{\kappa _4} - {\kappa _6}{n_2}{\kappa _3}} \over {{\kappa _3}{\kappa _1}}},{w_3} = {{{\kappa _2}{\kappa _5}{n_3} - {l_3}{\kappa _4}{\kappa _3} - {\kappa _3}{n_3}{\kappa _6}} \over {{\kappa _3}{\kappa _1}}}.} \hfill \cr }

Substituting the parameter values of equation (23) into equation (19), we attain 24 $\begin{array}{l} f & = & {(l_{1} x - \frac{κ_{2} n_{1} y}{κ_{3}} + n_{1} z + \frac{t (- κ_{3} l_{1} κ_{4} + κ_{2} κ_{5} n_{1} - κ_{3} n_{1} κ_{6})}{κ_{3} κ_{1}} + δ_{1})}^{2} + \\ {(l_{2} x - \frac{κ_{2} n_{2} y}{κ_{3}} + n_{2} z + \frac{t (- l_{2} κ_{3} κ_{4} + κ_{2} n_{2} κ_{5} - κ_{6} n_{2} κ_{3})}{κ_{3} κ_{1}} + δ_{2})}^{2} + \\ e^{l_{3} x - \frac{κ_{2} n_{3} y}{κ_{3}} + n_{3} z + \frac{t (- l_{3} κ_{4} κ_{3} + κ_{2} κ_{5} n_{3} - κ_{3} n_{3} κ_{6})}{κ_{3} κ_{1}}} + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( {{l_1}x - {{{\kappa _2}{n_1}y} \over {{\kappa _3}}} + {n_1}z + {{t\left( { - {\kappa _3}{l_1}{\kappa _4} + {\kappa _2}{\kappa _5}{n_1} - {\kappa _3}{n_1}{\kappa _6}} \right)} \over {{\kappa _3}{\kappa _1}}} + {\delta _1}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {{{\left( {{l_2}x - {{{\kappa _2}{n_2}y} \over {{\kappa _3}}} + {n_2}z + {{t\left( { - {l_2}{\kappa _3}{\kappa _4} + {\kappa _2}{n_2}{\kappa _5} - {\kappa _6}{n_2}{\kappa _3}} \right)} \over {{\kappa _3}{\kappa _1}}} + {\delta _2}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {{{\rm{e}}^{{l_3}x - {{{\kappa _2}{n_3}y} \over {{\kappa _3}}} + {n_3}z + {{t\left( { - {l_3}{\kappa _4}{\kappa _3} + {\kappa _2}{\kappa _5}{n_3} - {\kappa _3}{n_3}{\kappa _6}} \right)} \over {{\kappa _3}{\kappa _1}}}}} + \eta .} \hfill \cr }

Substituting the expression (24) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which reveals the collision of the single-lump and solitary-wave. Taking the specific value of the free parameters: l₁ = l₃ = 1, l₂ = –1, n₁ = –3, n₂ = n₃ = 1, δ₁ = δ₂ = 1, κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –1, κ₅ = –1, κ₆ = 1, and η = 1. The specific wave solution of the governing equation (1) is 25 $\begin{array}{l} u & = & \frac{4 + 2 e^{ϕ_{2}}}{e^{ϕ_{2}} + 45 t^{2} + 6 (x + 7 y - 7 z + 1) t + x^{2} + 2 (y - z - 1) x + 10 y^{2} + 4 (1 - 5 z) y + 10 z^{2} - 4 z + 3} - \\ \frac{2 {(2 x + 2 y - 2 z + 6 t - 2 + e^{ϕ_{2}})}^{2}}{{(e^{ϕ_{2}} + 45 t^{2} + 6 (x + 7 y - 7 z + 1) t + x^{2} + 2 (y - z - 1) x + 10 y^{2} + 4 (1 - 5 z) y + 10 z^{2} - 4 z + 3)}^{2}}, \end{array}$ \matrix{ u \hfill & = \hfill & {{{4 + 2{e^{{\phi _2}}}} \over {{e^{{\phi _2}}} + 45{t^2} + 6(x + 7y - 7z + 1)t + {x^2} + 2(y - z - 1)x + 10{y^2} + 4(1 - 5z)y + 10{z^2} - 4z + 3}} - } \hfill \cr {} \hfill & {} \hfill & {{{2{{\left( {2x + 2y - 2z + 6t - 2 + {e^{{\phi _2}}}} \right)}^2}} \over {{{\left( {{e^{{\phi _2}}} + 45{t^2} + 6(x + 7y - 7z + 1)t + {x^2} + 2(y - z - 1)x + 10{y^2} + 4(1 - 5z)y + 10{z^2} - 4z + 3} \right)}^2}}},} \hfill \cr } where ϕ₂ = x – y + z – t. Its propagation velocity along the xy-plane is shown in Fig. 4.

4.3

Collision of waves: lump wave and periodic wave

Assume a function that combines the sine and quadratic functions to examine the conflict of the lump and periodic wave solutions of the KDKK and BK equations (1). For example, 26 $f = {(l_{1} x + m_{1} y + n_{1} z + w_{1} t + δ_{1})}^{2} + {(l_{2} x + m_{2} y + n_{2} z + w_{2} t + δ_{2})}^{2} + s i n (l_{3} x + m_{3} y + n_{3} z + w_{3} t) + η,$ f = {\left( {{l_1}x + {m_1}y + {n_1}z + {w_1}t + {\delta _1}} \right)^2} + {\left( {{l_2}x + {m_2}y + {n_2}z + {w_2}t + {\delta _2}} \right)^2} + \sin \left( {{l_3}x + {m_3}y + {n_3}z + {w_3}t} \right) + \eta , where l_r, m_r, n_r, w_r, δ₁, δ₂, and η are free parameters with r = 1,2,3 to be determined. Substituting equation (26) into equation (11) and applying the same procedure as before. A system of algebraic equations involving l_r, m_r, n_r, w_r, δ₁, δ₂, and η is obtained, where 1 ≤ r ≤ 3. Solving the system of equations using the symbolic software Maple, we obtain the values of some parameters:

Case I

l_{1} = 0, l_{2} = - \frac{w_{2} κ_{1} + m_{2} κ_{5} + n_{2} κ_{6}}{κ_{4}}, l_{3} = 0, m_{1} = 0, n_{1} = 0, n_{2} = - \frac{κ_{3} m_{2}}{κ_{2}}, w_{1} = 0, w_{3} = - \frac{m_{3} κ_{5} + n_{3} κ_{6}}{κ_{1}} .

{l_1} = 0,{l_2} = - {{{w_2}{\kappa _1} + {m_2}{\kappa _5} + {n_2}{\kappa _6}} \over {{\kappa _4}}},{l_3} = 0,{m_1} = 0,{n_1} = 0,{n_2} = - {{{\kappa _3}{m_2}} \over {{\kappa _2}}},{w_1} = 0,{w_3} = - {{{m_3}{\kappa _5} + {n_3}{\kappa _6}} \over {{\kappa _1}}}.

Substituting the parameter values of equation (27) into equation (26), we attain 28 $f = δ_{1}^{2} + {(- \frac{(m_{2} κ_{5} - \frac{κ_{3} m_{2} κ_{6}}{κ_{2}} + w_{2} κ_{1}) x}{κ_{4}} + m_{2} y - \frac{κ_{3} m_{2} z}{κ_{2}} + w_{2} t + δ_{2})}^{2} - \sin (- m_{3} y - n_{3} z + \frac{t (m_{3} κ_{5} + n_{3} κ_{6})}{κ_{1}}) + η .$ f = \delta _1^2 + {\left( { - {{\left( {{m_2}{\kappa _5} - {{{\kappa _3}{m_2}{\kappa _6}} \over {{\kappa _2}}} + {w_2}{\kappa _1}} \right)x} \over {{\kappa _4}}} + {m_2}y - {{{\kappa _3}{m_2}z} \over {{\kappa _2}}} + {w_2}t + {\delta _2}} \right)^2} - \sin \left( { - {m_3}y - {n_3}z + {{t\left( {{m_3}{\kappa _5} + {n_3}{\kappa _6}} \right)} \over {{\kappa _1}}}} \right) + \eta .

Substituting the expression (28) into equation (6) yields the appropriate solution to equation (1), which reveals the conflict between the lump wave and periodic wave. Choosing the particular value of the free parameters: m₂ = m₃ = n₃ = w₂ = 1, δ₁ = δ₂ = 1, κ₁ = 1, κ₂ = –1, κ₃ = 1, κ₄ = –3, κ₅ = κ₆ = 1, and η = 1. The particular wave solution of the KDKK and BK equations (1) is 29 $u = - \frac{\sin (2 t - F_{1} + 1) + t^{2} + 2 (x + F_{1}) t + x^{2} + 2 F_{1} x + y^{2} + 2 (1 + z) y + z^{2} + 2 z - 1}{{(- \frac{1}{2} \sin (2 t - F_{1} + 1) + \frac{1}{2} (x^{2} + y^{2} + z^{2} + t^{2} + 3) + (x + F_{1}) t + F_{1} x + (1 + z) y + z)}^{2}},$ u = - {{\sin \left( {2t - {F_1} + 1} \right) + {t^2} + 2\left( {x + {F_1}} \right)t + {x^2} + 2{F_1}x + {y^2} + 2(1 + z)y + {z^2} + 2z - 1} \over {{{\left( { - {1 \over 2}\sin \left( {2t - {F_1} + 1} \right) + {1 \over 2}\left( {{x^2} + {y^2} + {z^2} + {t^2} + 3} \right) + \left( {x + {F_1}} \right)t + {F_1}x + (1 + z)y + z} \right)}^2}}}, where F₁ = 1 + y + z. Its propagation velocity along the xy-plane, xz-plane, and xt-plane are shown in Fig. 5.

Case II

\begin{array}{l} l_{1} & = & 0, l_{2} = - \frac{m_{1} w_{2} κ_{5} - m_{2} w_{1} κ_{5} + n_{1} κ_{6} w_{2} - n_{2} κ_{6} w_{1}}{κ_{4} w_{1}}, l_{3} = 0, \\ n_{2} & = & - \frac{m_{2} κ_{3}}{κ_{2}}, w_{1} = - \frac{m_{1} κ_{5} + n_{1} κ_{6}}{κ_{1}}, w_{3} = \frac{(m_{3} κ_{5} + n_{3} κ_{6}) w_{1}}{m_{1} κ_{5} + n_{1} κ_{6}} . \end{array}

\matrix{ {{l_1}} \hfill & = \hfill & {0,{l_2} = - {{{m_1}{w_2}{\kappa _5} - {m_2}{w_1}{\kappa _5} + {n_1}{\kappa _6}{w_2} - {n_2}{\kappa _6}{w_1}} \over {{\kappa _4}{w_1}}},{l_3} = 0,} \hfill \cr {{n_2}} \hfill & = \hfill & { - {{{m_2}{\kappa _3}} \over {{\kappa _2}}},{w_1} = - {{{m_1}{\kappa _5} + {n_1}{\kappa _6}} \over {{\kappa _1}}},{w_3} = {{\left( {{m_3}{\kappa _5} + {n_3}{\kappa _6}} \right){w_1}} \over {{m_1}{\kappa _5} + {n_1}{\kappa _6}}}.} \hfill \cr }

Substituting the parameter values of equation (30) into equation (26), we attain 31 $\begin{array}{l} f & = & {(m_{1} y + n_{1} z - \frac{t (m_{1} κ_{5} + n_{1} κ_{6})}{κ_{1}} + δ_{1})}^{2} + (m_{2} y - \frac{κ_{3} m_{2} z}{κ_{2}} + w_{2} t + δ_{2} + \\ {\frac{(m_{1} w_{2} κ_{5} + \frac{m_{2} (m_{1} κ_{5} + n_{1} κ_{6}) κ_{5}}{κ_{1}} + n_{1} κ_{6} w_{2} - \frac{κ_{3} m_{2} (m_{1} κ_{5} + n_{1} κ_{6}) κ_{6}}{κ_{2} κ_{1}}) κ_{1} x}{κ_{4} (m_{1} κ_{5} + n_{1} κ_{6})})}^{2} - \\ \sin (- m_{3} y - n_{3} z + \frac{t (m_{3} κ_{5} + n_{3} κ_{6})}{κ_{1}}) + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( {{m_1}y + {n_1}z - {{t\left( {{m_1}{\kappa _5} + {n_1}{\kappa _6}} \right)} \over {{\kappa _1}}} + {\delta _1}} \right)}^2} + \left( {{m_2}y - {{{\kappa _3}{m_2}z} \over {{\kappa _2}}} + {w_2}t + {\delta _2} + } \right.} \hfill \cr {} \hfill & {} \hfill & {{{\left. {{{\left( {{m_1}{w_2}{\kappa _5} + {{{m_2}\left( {{m_1}{\kappa _5} + {n_1}{\kappa _6}} \right){\kappa _5}} \over {{\kappa _1}}} + {n_1}{\kappa _6}{w_2} - {{{\kappa _3}{m_2}\left( {{m_1}{\kappa _5} + {n_1}{\kappa _6}} \right){\kappa _6}} \over {{\kappa _2}{\kappa _1}}}} \right){\kappa _1}x} \over {{\kappa _4}\left( {{m_1}{\kappa _5} + {n_1}{\kappa _6}} \right)}}} \right)}^2} - } \hfill \cr {} \hfill & {} \hfill & {\sin \left( { - {m_3}y - {n_3}z + {{t\left( {{m_3}{\kappa _5} + {n_3}{\kappa _6}} \right)} \over {{\kappa _1}}}} \right) + \eta .} \hfill \cr }

Substituting the expression (31) into equation (6) yields the appropriate solution to equation (1), which reveals the collision of periodic and lump wave. Choosing the particular value of the free parameters: κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –1, κ₅ = 1, κ₆ = 1, m₁ = i, m₂ = m₃ = 1, n₁ = n₃ = 1, w₂ = 1, δ₁ = δ₂ = 1, and $η = - \frac{1}{2}$ \eta = - {1 \over 2}, The particular wave solution of the KDKK and BK equations (1) is 32 $\begin{array}{l} u & = & \frac{8}{2 {(1 + i y + z - (1 + i) t)}^{2} + 2 {(1 - x + y - z + t)}^{2} - 2 \sin (2 t - y - z) - 1} - \\ \frac{8 {(1 - x + y - z + t)}^{2}}{{({(1 + i y + z - (1 + i) t)}^{2} + {(1 - x + y - z + t)}^{2} - \frac{1}{2} - \sin (2 t - y - z))}^{2}} \end{array} .$ \matrix{ u \hfill & = \hfill & {{8 \over {2{{(1 + iy + z - (1 + i)t)}^2} + 2{{(1 - x + y - z + t)}^2} - 2\sin (2t - y - z) - 1}} - } \hfill \cr {} \hfill & {} \hfill & {{{8{{(1 - x + y - z + t)}^2}} \over {{{\left( {{{(1 + iy + z - (1 + i)t)}^2} + {{(1 - x + y - z + t)}^2} - {1 \over 2} - \sin (2t - y - z)} \right)}^2}}}} \hfill \cr } .

The interaction of the periodic and lump wave solutions of equation (32) over time is shown in Fig. 6 through contour plots, 2D profiles, and 3D profiles.

Case III

\begin{array}{l} l_{1} = 0, l_{2} = - \frac{κ_{1} m_{1} w_{2} + n_{2} κ_{6} m_{1} - n_{1} m_{2} κ_{6}}{κ_{4} m_{1}}, l_{3} = 0, m_{1} = - \frac{n_{1} κ_{6}}{κ_{5}}, \\ n_{2} = - \frac{κ_{3} m_{2}}{κ_{2}}, w_{1} = 0, w_{3} = - \frac{κ_{6} (m_{1} n_{3} - m_{3} n_{1})}{κ_{1} m_{1}} . \end{array}

\matrix{ {{l_1} = 0,{l_2} = - {{{\kappa _1}{m_1}{w_2} + {n_2}{\kappa _6}{m_1} - {n_1}{m_2}{\kappa _6}} \over {{\kappa _4}{m_1}}},{l_3} = 0,{m_1} = - {{{n_1}{\kappa _6}} \over {{\kappa _5}}},} \hfill \cr {{n_2} = - {{{\kappa _3}{m_2}} \over {{\kappa _2}}},{w_1} = 0,{w_3} = - {{{\kappa _6}\left( {{m_1}{n_3} - {m_3}{n_1}} \right)} \over {{\kappa _1}{m_1}}}.} \hfill \cr }

Substituting the parameter values of equation (33) into equation (26), we attain 34 $\begin{array}{l} f & = & {(- \frac{n_{1} κ_{6} y}{κ_{5}} + n_{1} z + δ_{1})}^{2} + {(\frac{(- \frac{κ_{1} n_{1} κ_{6} w_{2}}{κ_{5}} + \frac{n_{1} κ_{6}^{2} κ_{3} m_{2}}{κ_{5} κ_{2}} - n_{1} m_{2} κ_{6}) κ_{5} x}{κ_{4} n_{1} κ_{6}} + m_{2} y - \frac{κ_{3} m_{2} z}{κ_{2}} + w_{2} t + δ_{2})}^{2} + \\ \sin (m_{3} y + n_{3} z + \frac{t (- \frac{n_{1} κ_{6} n_{3}}{κ_{5}} - m_{3} n_{1}) κ_{5}}{κ_{1} n_{1}}) + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( { - {{{n_1}{\kappa _6}y} \over {{\kappa _5}}} + {n_1}z + {\delta _1}} \right)}^2} + {{\left( {{{\left( { - {{{\kappa _1}{n_1}{\kappa _6}{w_2}} \over {{\kappa _5}}} + {{{n_1}\kappa _6^2{\kappa _3}{m_2}} \over {{\kappa _5}{\kappa _2}}} - {n_1}{m_2}{\kappa _6}} \right){\kappa _5}x} \over {{\kappa _4}{n_1}{\kappa _6}}} + {m_2}y - {{{\kappa _3}{m_2}z} \over {{\kappa _2}}} + {w_2}t + {\delta _2}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {\sin \left( {{m_3}y + {n_3}z + {{t\left( { - {{{n_1}{\kappa _6}{n_3}} \over {{\kappa _5}}} - {m_3}{n_1}} \right){\kappa _5}} \over {{\kappa _1}{n_1}}}} \right) + \eta .} \hfill \cr }

Substituting the expression (34) into equation (6) yields the appropriate solutions to equation (1), which reveals the collision between periodic and lump wave. Choosing the particular value of the free parameters m₂ = m₃ = 1, n₁ = i, n₃ = –1, w₂ = 1, δ₁ = δ₂ = 1, κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –2, κ₅ = –1, κ₆ = 1, and η = 1. The particular wave solution of the KDKK and BK equations (1) is 35 $u = \frac{1}{{(1 + i y + i z)}^{2} + F_{2}^{2} + \sin (2 t + y - z) + 1} - \frac{2 F_{2}^{2}}{{({(1 + i y + i z)}^{2} + F_{2}^{2} + \sin (2 t + y - z) + 1)}^{2}},$ u = {1 \over {{{(1 + iy + iz)}^2} + F_2^2 + \sin (2t + y - z) + 1}} - {{2F_2^2} \over {{{\left( {{{(1 + iy + iz)}^2} + F_2^2 + \sin (2t + y - z) + 1} \right)}^2}}}, where $F_{2} = 1 - \frac{x}{2} + y - z + t$ {F_2} = 1 - {x \over 2} + y - z + t. Its propagation velocity is shown in Fig. 7.

4.4

Collision of waves: solitary, periodic, and lump waves

Consider a function that combines exponential, quadratic, and cosine functions to analyze the conflict that arises between solitary, periodic, and lump wave solutions of the KDKK and BK equation (1), that is, 36 $f = {(l_{1} x + m_{1} y + n_{1} z + w_{1} t + δ_{1})}^{2} + {(l_{2} x + m_{2} y + n_{2} z + w_{2} t + δ_{2})}^{2} + c o s (l_{3} x + m_{3} y + n_{3} z + w_{3} t) + e x p (l_{3} x + m_{3} y + n_{3} z + w_{3} t) + η,$ f = {\left( {{l_1}x + {m_1}y + {n_1}z + {w_1}t + {\delta _1}} \right)^2} + {\left( {{l_2}x + {m_2}y + {n_2}z + {w_2}t + {\delta _2}} \right)^2} + \cos \left( {{l_3}x + {m_3}y + {n_3}z + {w_3}t} \right) + \exp \left( {{l_3}x + {m_3}y + {n_3}z + {w_3}t} \right) + \eta , where l_r, m_r, n_r, w_r, δ₁, δ₂, and η are free parameters with r = 1,2,3 to be determined. Substituting equation (36) into equation (11) and following the same procedure as before. A system of algebraic equations consisting of l_r, m_r, n_r, w_r, δ₁, δ₂, and η with 1 ≤ r ≤ 3 is obtained. Solving the system of equations using the symbolic software Maple, we obtain the values of some parameters:

Case I

\begin{array}{l} l_{1} & = & 0, l_{2} = - \frac{κ_{1} κ_{2} n_{2} w_{1} + w_{2} κ_{1} κ_{3} m_{1} + κ_{2} n_{1} n_{2} κ_{6} + n_{2} κ_{6} κ_{3} m_{1}}{κ_{4} κ_{3} m_{1}}, l_{3} = 0, l_{4} = 0, \\ m_{1} & = & - \frac{w_{1} κ_{1} + n_{1} κ_{6}}{κ_{5}}, m_{2} = - \frac{κ_{2} n_{2}}{κ_{3}}, n_{3} = - \frac{w_{3} κ_{1} m_{1} - κ_{1} m_{3} w_{1} - m_{3} n_{1} κ_{6}}{m_{1} κ_{6}}, \\ n_{4} & = & - \frac{w_{4} κ_{1} m_{1} - κ_{1} m_{4} w_{1} - m_{4} n_{1} κ_{6}}{m_{1} κ_{6}} . \end{array}

\matrix{ {{l_1}} \hfill & = \hfill & {0,{l_2} = - {{{\kappa _1}{\kappa _2}{n_2}{w_1} + {w_2}{\kappa _1}{\kappa _3}{m_1} + {\kappa _2}{n_1}{n_2}{\kappa _6} + {n_2}{\kappa _6}{\kappa _3}{m_1}} \over {{\kappa _4}{\kappa _3}{m_1}}},{l_3} = 0,{l_4} = 0,} \hfill \cr {{m_1}} \hfill & = \hfill & { - {{{w_1}{\kappa _1} + {n_1}{\kappa _6}} \over {{\kappa _5}}},{m_2} = - {{{\kappa _2}{n_2}} \over {{\kappa _3}}},{n_3} = - {{{w_3}{\kappa _1}{m_1} - {\kappa _1}{m_3}{w_1} - {m_3}{n_1}{\kappa _6}} \over {{m_1}{\kappa _6}}},} \hfill \cr {{n_4}} \hfill & = \hfill & { - {{{w_4}{\kappa _1}{m_1} - {\kappa _1}{m_4}{w_1} - {m_4}{n_1}{\kappa _6}} \over {{m_1}{\kappa _6}}}.} \hfill \cr }

Substituting the parameter values of equation (37) into equation (36), we attain 38 $\begin{array}{l} f & = & {(- \frac{(κ_{1} w_{1} + n_{1} κ_{6}) y}{κ_{5}} + n_{1} z + w_{1} t + δ_{1})}^{2} + (- \frac{κ_{2} n_{2} y}{κ_{3}} + n_{2} z + w_{2} t + δ_{2} + \\ {\frac{(κ_{1} κ_{2} n_{2} w_{1} - \frac{κ_{1} κ_{3} (κ_{1} w_{1} + n_{1} κ_{6}) w_{2}}{κ_{5}} + κ_{2} n_{1} n_{2} κ_{6} - \frac{κ_{3} (κ_{1} w_{1} + n_{1} κ_{6}) n_{2} κ_{6}}{κ_{5}}) κ_{5} x}{κ_{4} κ_{3} (κ_{1} w_{1} + n_{1} κ_{6})})}^{2} + \\ \cos (- m_{3} y - \frac{(- \frac{κ_{1} (κ_{1} w_{1} + n_{1} κ_{6}) w_{3}}{κ_{5}} - κ_{1} m_{3} w_{1} - m_{3} n_{1} κ_{6}) κ_{5} z}{(κ_{1} w_{1} + n_{1} κ_{6}) κ_{6}} - w_{3} t) + \\ e^{m_{4} y + \frac{(- \frac{κ_{1} (κ_{1} w_{1} + n_{1} κ_{6}) w_{4}}{κ_{5}} - κ_{1} m_{4} w_{1} - m_{4} n_{1} κ_{6}) κ_{5} z}{(κ_{1} w_{1} + n_{1} κ_{6}) κ_{6}} + w_{4} t} + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( { - {{\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right)y} \over {{\kappa _5}}} + {n_1}z + {w_1}t + {\delta _1}} \right)}^2} + \left( { - {{{\kappa _2}{n_2}y} \over {{\kappa _3}}} + {n_2}z + {w_2}t + {\delta _2} + } \right.} \hfill \cr {} \hfill & {} \hfill & {{{\left. {{{\left( {{\kappa _1}{\kappa _2}{n_2}{w_1} - {{{\kappa _1}{\kappa _3}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){w_2}} \over {{\kappa _5}}} + {\kappa _2}{n_1}{n_2}{\kappa _6} - {{{\kappa _3}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){n_2}{\kappa _6}} \over {{\kappa _5}}}} \right){\kappa _5}x} \over {{\kappa _4}{\kappa _3}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right)}}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {\cos \left( { - {m_3}y - {{\left( { - {{{\kappa _1}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){w_3}} \over {{\kappa _5}}} - {\kappa _1}{m_3}{w_1} - {m_3}{n_1}{\kappa _6}} \right){\kappa _5}z} \over {\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){\kappa _6}}} - {w_3}t} \right) + } \hfill \cr {} \hfill & {} \hfill & {{{\rm{e}}^{{m_4}y + {{\left( { - {{{\kappa _1}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){w_4}} \over {{\kappa _5}}} - {\kappa _1}{m_4}{w_1} - {m_4}{n_1}{\kappa _6}} \right){\kappa _5}z} \over {\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){\kappa _6}}} + {w_4}t}} + \eta .} \hfill \cr }

Substituting the expression (38) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which reveals the collision of solitary, periodic, and lump waves. Taking the specific value of the parameters: m₃ = –0.2, m₄ = 0.4, n₁ = 0.3, n₂ = –i, w₁ = 0.1, w₂ = 0.1, w₃ = 0.2, w₄ = 0.5, δ₁ = δ₂ = 1.5, κ₁ = 1, κ₂ = 0.5, κ₃ = 1, κ₄ = –4.5, κ₅ = –1, κ₆ = 1, and η = –0.6. The specific wave solution of the governing equation (1) is 39 $\begin{array}{l} u & = & - \frac{0.4 + 0.06 i}{{(0.4 y + 0.3 z + 0.1 t + 1.5)}^{2} + P^{2} - 0.6 + \cos (- 0.2 y - 0.4 z + 0.2 t) + e^{0.4 y - 0.1 z + 0.5 t}} + \\ \frac{(0.9 + 0.12 i) {((0.02 - 0.3 i) x + 0.5 i y - i z + 0.1 t + 1.5)}^{2}}{{({(0.4 y + 0.3 z + 0.1 t + 1.5)}^{2} + P^{2} - 0.6 + \cos (- 0.2 y - 0.4 z + 0.2 t) + e^{0.4 y - 0.1 z + 0.5 t})}^{2}}, \end{array}$ \matrix{ u \hfill & = \hfill & { - {{0.4 + 0.06i} \over {{{(0.4y + 0.3z + 0.1t + 1.5)}^2} + {P^2} - 0.6 + \cos ( - 0.2y - 0.4z + 0.2t) + {{\rm{e}}^{0.4y - 0.1z + 0.5t}}}} + } \hfill \cr {} \hfill & {} \hfill & {{{(0.9 + 0.12i){{((0.02 - 0.3i)x + 0.5iy - iz + 0.1t + 1.5)}^2}} \over {{{\left( {{{(0.4y + 0.3z + 0.1t + 1.5)}^2} + {P^2} - 0.6 + \cos ( - 0.2y - 0.4z + 0.2t) + {{\rm{e}}^{0.4y - 0.1z + 0.5t}}} \right)}^2}}},} \hfill \cr } where P = (0.02 – 0.3i)x + 0.5iy – iz + 0.1t + 1.5. Fig. 8 illustrates the interaction of solitary, periodic, and lump wave solutions of equation (39) through contour plots, 2D profiles, and 3D profiles.

Case II

\begin{array}{l} l_{1} = l_{3} = l_{4} = 0, l_{2} = - \frac{w_{2} κ_{1}}{κ_{4}}, m_{1} = m_{2} = 0, n_{1} = - \frac{w_{1} κ_{1}}{κ_{6}}, \\ n_{2} = 0, n_{3} = - \frac{w_{3} κ_{1} + m_{3} κ_{5}}{κ_{6}}, n_{4} = - \frac{w_{4} κ_{1} + m_{4} κ_{5}}{κ_{6}} . \end{array}

\matrix{ {{l_1} = {l_3} = {l_4} = 0,{l_2} = - {{{w_2}{\kappa _1}} \over {{\kappa _4}}},{m_1} = {m_2} = 0,{n_1} = - {{{w_1}{\kappa _1}} \over {{\kappa _6}}},} \hfill \cr {{n_2} = 0,{n_3} = - {{{w_3}{\kappa _1} + {m_3}{\kappa _5}} \over {{\kappa _6}}},{n_4} = - {{{w_4}{\kappa _1} + {m_4}{\kappa _5}} \over {{\kappa _6}}}.} \hfill \cr }

Substituting the parameter values of equation (40) into equation (36), we attain 41 $\begin{array}{l} f & = & {(- \frac{w_{1} κ_{1} z}{κ_{6}} + w_{1} t + δ_{1})}^{2} + {(- \frac{w_{2} κ_{1} x}{κ_{4}} + w_{2} t + δ_{2})}^{2} - \\ \sin (- m_{3} y + \frac{(w_{3} κ_{1} + m_{3} κ_{5}) z}{κ_{6}} - w_{3} t) + e^{m_{4} y - \frac{(w_{4} κ_{1} + m_{4} κ_{5}) z}{κ_{6}} + w_{4} t} + η . \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( { - {{{w_1}{\kappa _1}z} \over {{\kappa _6}}} + {w_1}t + {\delta _1}} \right)}^2} + {{\left( { - {{{w_2}{\kappa _1}x} \over {{\kappa _4}}} + {w_2}t + {\delta _2}} \right)}^2} - } \hfill \cr {} \hfill & {} \hfill & {\sin \left( { - {m_3}y + {{\left( {{w_3}{\kappa _1} + {m_3}{\kappa _5}} \right)z} \over {{\kappa _6}}} - {w_3}t} \right) + {{\rm{e}}^{{m_4}y - {{\left( {{w_4}{\kappa _1} + {m_4}{\kappa _5}} \right)z} \over {{\kappa _6}}} + {w_4}t}} + \eta .} \hfill \cr }

Substituting the expression (41) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which reveals the collision of solitary, periodic, and lump waves. Taking the specific value of the free parameters: m₃ = –1, m₄ = 0.5, w₁ = 0.8, w₂ = 0.1, w₃ = 0.6, w₄ = 0.3, δ₁ = δ₂ = 1.5, η = –0.3, κ₁ = 2, κ₂ = 0.2, κ₃ = 1, κ₄ = –2, κ₅ = –1, and κ₆ = 1. The specific wave solution of the governing equation (1) is 42 $\begin{array}{l} u & = & \frac{0.04}{{(- 1.6 z + 0.8 t + 1.5)}^{2} + {(0.1 x + 0.1 t + 1.5)}^{2} - 0.3 + \sin (- y - 2.2 z + 0.6 t) + e^{0.5 y - 0.1 z + 0.3 t}} - \\ \frac{0.08 {(0.1 x + 0.1 t + 1.5)}^{2}}{{({(- 1.6 z + 0.8 t + 1.5)}^{2} + {(0.1 x + 0.1 t + 1.5)}^{2} - 0.3 + \sin (- y - 2.2 z + 0.6 t) + e^{0.5 y - 0.1 z + 0.3 t})}^{2}} . \end{array}$ \matrix{ u \hfill & = \hfill & {{{0.04} \over {{{( - 1.6z + 0.8t + 1.5)}^2} + {{(0.1x + 0.1t + 1.5)}^2} - 0.3 + \sin ( - y - 2.2z + 0.6t) + {{\rm{e}}^{0.5y - 0.1z + 0.3t}}}} - } \hfill \cr {} \hfill & {} \hfill & {{{0.08{{(0.1x + 0.1t + 1.5)}^2}} \over {{{\left( {{{( - 1.6z + 0.8t + 1.5)}^2} + {{(0.1x + 0.1t + 1.5)}^2} - 0.3 + \sin ( - y - 2.2z + 0.6t) + {{\rm{e}}^{0.5y - 0.1z + 0.3t}}} \right)}^2}}}.} \hfill \cr }

Its propagation velocity along the xy-plane is shown in Fig. 9.

4.5

Collision of waves: lump and breather waves

Consider a function that combines cosine, the double exponential, and quadratic functions to test the collision of lump and breather wave solutions of KDKK and BK equations (1), that is, 43 $\begin{array}{l} f & = & {(l_{1} x + m_{1} y + n_{1} z + w_{1} t + δ_{1})}^{2} + {(l_{2} x + m_{2} y + n_{2} z + w_{2} t + δ_{2})}^{2} + c o s (l_{3} x + m_{3} y + n_{3} z + w_{3} t + δ_{3}) + \\ e x p (l_{4} x + m_{4} y + n_{4} z + w_{4} t + δ_{4}) + e x p (- l_{4} x - m_{4} y - n_{4} z - w_{4} t - δ_{4}) + η, \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( {{l_1}x + {m_1}y + {n_1}z + {w_1}t + {\delta _1}} \right)}^2} + {{\left( {{l_2}x + {m_2}y + {n_2}z + {w_2}t + {\delta _2}} \right)}^2} + \cos \left( {{l_3}x + {m_3}y + {n_3}z + {w_3}t + {\delta _3}} \right) + } \hfill \cr {} \hfill & {} \hfill & {\exp \left( {{l_4}x + {m_4}y + {n_4}z + {w_4}t + {\delta _4}} \right) + \exp \left( { - {l_4}x - {m_4}y - {n_4}z - {w_4}t - {\delta _4}} \right) + \eta ,} \hfill \cr } where l_r, m_r, n_r, w_r, δ_r, and η are free parameters with r = 1,2,3,4 to be determined. Substituting equation (43) into equation (11) and following the same procedure as before. Then, a system of algebraic equations involving l_r, m_r, n_r, w_r, δ_r, and η is obtained, where 1 ≤ r ≤ 4. Solving the system of equations using the symbolic software Maple, we obtain the values of some parameters: 44 $\begin{array}{l} l_{1} = 0, l_{2} = - \frac{κ_{1} κ_{2} n_{2} w_{1} + w_{2} κ_{1} κ_{3} m_{1} + κ_{2} n_{1} n_{2} κ_{6} + n_{2} κ_{6} κ_{3} m_{1}}{κ_{4} κ_{3} m_{1}}, l_{3} = 0, l_{4} = 0, \\ m_{1} = - \frac{w_{1} κ_{1} + n_{1} κ_{6}}{κ_{5}}, m_{2} = - \frac{κ_{2} n_{2}}{κ_{3}}, n_{3} = - \frac{w_{3} κ_{1} m_{1} - κ_{1} m_{3} w_{1} - m_{3} n_{1} κ_{6}}{m_{1} κ_{6}}, \\ n_{4} = - \frac{w_{4} κ_{1} m_{1} - κ_{1} m_{4} w_{1} - m_{4} n_{1} κ_{6}}{m_{1} κ_{6}} . \end{array}$ \matrix{ {{l_1} = 0,{l_2} = - {{{\kappa _1}{\kappa _2}{n_2}{w_1} + {w_2}{\kappa _1}{\kappa _3}{m_1} + {\kappa _2}{n_1}{n_2}{\kappa _6} + {n_2}{\kappa _6}{\kappa _3}{m_1}} \over {{\kappa _4}{\kappa _3}{m_1}}},{l_3} = 0,{l_4} = 0,} \hfill \cr {{m_1} = - {{{w_1}{\kappa _1} + {n_1}{\kappa _6}} \over {{\kappa _5}}},{m_2} = - {{{\kappa _2}{n_2}} \over {{\kappa _3}}},{n_3} = - {{{w_3}{\kappa _1}{m_1} - {\kappa _1}{m_3}{w_1} - {m_3}{n_1}{\kappa _6}} \over {{m_1}{\kappa _6}}},} \hfill \cr {{n_4} = - {{{w_4}{\kappa _1}{m_1} - {\kappa _1}{m_4}{w_1} - {m_4}{n_1}{\kappa _6}} \over {{m_1}{\kappa _6}}}.} \hfill \cr }

Substituting the parameter values of equation (44) into equation (43), we attain 45 $\begin{array}{l} f & = & {(- \frac{(κ_{1} w_{1} + n_{1} κ_{6}) y}{κ_{5}} + n_{1} z + w_{1} t + δ_{1})}^{2} + (- \frac{κ_{2} n_{2} y}{κ_{3}} + n_{2} z + w_{2} t + δ_{2} + \\ {\frac{(κ_{1} κ_{2} n_{2} w_{1} - \frac{κ_{1} κ_{3} (κ_{1} w_{1} + n_{1} κ_{6}) w_{2}}{κ_{5}} + κ_{2} n_{1} n_{2} κ_{6} - \frac{κ_{3} (κ_{1} w_{1} + n_{1} κ_{6}) n_{2} κ_{6}}{κ_{5}}) κ_{5} x}{κ_{4} κ_{3} (κ_{1} w_{1} + n_{1} κ_{6})})}^{2} + \\ \cos (- m_{3} y - \frac{(- \frac{κ_{1} (κ_{1} w_{1} + n_{1} κ_{6}) w_{3}}{κ_{5}} - κ_{1} m_{3} w_{1} - m_{3} n_{1} κ_{6}) κ_{5} z}{(κ_{1} w_{1} + n_{1} κ_{6}) κ_{6}} - w_{3} t - δ_{3}) + \\ e^{m_{4} y + S z + w_{4} t + δ_{4}} + e^{- m_{4} y - S z - w_{4} t - δ_{4}} + η, \end{array}$ \matrix{ f \hfill & = \hfill & {{{\left( { - {{\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right)y} \over {{\kappa _5}}} + {n_1}z + {w_1}t + {\delta _1}} \right)}^2} + \left( { - {{{\kappa _2}{n_2}y} \over {{\kappa _3}}} + {n_2}z + {w_2}t + {\delta _2} + } \right.} \hfill \cr {} \hfill & {} \hfill & {{{\left. {{{\left( {{\kappa _1}{\kappa _2}{n_2}{w_1} - {{{\kappa _1}{\kappa _3}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){w_2}} \over {{\kappa _5}}} + {\kappa _2}{n_1}{n_2}{\kappa _6} - {{{\kappa _3}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){n_2}{\kappa _6}} \over {{\kappa _5}}}} \right){\kappa _5}x} \over {{\kappa _4}{\kappa _3}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right)}}} \right)}^2} + } \hfill \cr {} \hfill & {} \hfill & {\cos \left( { - {m_3}y - {{\left( { - {{{\kappa _1}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){w_3}} \over {{\kappa _5}}} - {\kappa _1}{m_3}{w_1} - {m_3}{n_1}{\kappa _6}} \right){\kappa _5}z} \over {\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){\kappa _6}}} - {w_3}t - {\delta _3}} \right) + } \hfill \cr {} \hfill & {} \hfill & {{e^{{m_4}y + Sz + {w_4}t + {\delta _4}}} + {e^{ - {m_4}y - Sz - {w_4}t - {\delta _4}}} + \eta ,} \hfill \cr } where $S = \frac{(- \frac{κ_{1} (κ_{1} w_{1} + n_{1} κ_{6}) w_{4}}{κ_{5}} - κ_{1} m_{4} w_{1} - m_{4} n_{1} κ_{6}) κ_{5}}{(κ_{1} w_{1} + n_{1} κ_{6}) κ_{6}} .$ S = {{\left( { - {{{\kappa _1}\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){w_4}} \over {{\kappa _5}}} - {\kappa _1}{m_4}{w_1} - {m_4}{n_1}{\kappa _6}} \right){\kappa _5}} \over {\left( {{\kappa _1}{w_1} + {n_1}{\kappa _6}} \right){\kappa _6}}}.

Substituting the expression (45) into equation (6) yields the appropriate solutions to the KDKK and BK equations (1), which reveals the collision between the breather and lump wave. Choosing the particular value of the free parameters: m₃ = –0.2, m₄ = 0.4, n₁ = 0.3, n₂ = –i, w₁ = 0.1, w₂ = 0.2, w₃ = 0.3, w₄ = 0.4, δ₁ = δ₂ = δ₃ = δ₄ = 1.5, η = –0.6, κ₁ = 0.2, κ₂ = 0.4, κ₃ = 0.5, κ₄ = –3, κ₅ = –1, and κ₆ = 1. The specific wave solution of the governing equation (1) is 46 $\begin{array}{l} u & = & - \frac{1.4 + 0.06 i}{{(0.3 y + 0.3 z + 0.1 t + 1.5)}^{2} + Q^{2} - 0.6 + \cos (- 0.2 y - 0.26 z + 0.3 t + 1.5) + e^{ω} + e^{- ω}} + \\ \frac{(2.88 + 0.13 i) Q^{2}}{{({(0.3 y + 0.3 z + 0.1 t + 1.5)}^{2} + Q^{2} - 0.6 + \cos (- 0.2 y - 0.26 z + 0.3 t + 1.5) + e^{ω} + e^{- ω})}^{2}}, \end{array}$ \matrix{ u \hfill & = \hfill & { - {{1.4 + 0.06i} \over {{{(0.3y + 0.3z + 0.1t + 1.5)}^2} + {Q^2} - 0.6 + \cos ( - 0.2y - 0.26z + 0.3t + 1.5) + {e^\omega } + {e^{ - \omega }}}} + } \hfill \cr {} \hfill & {} \hfill & {{{(2.88 + 0.13i){Q^2}} \over {{{\left( {{{(0.3y + 0.3z + 0.1t + 1.5)}^2} + {Q^2} - 0.6 + \cos ( - 0.2y - 0.26z + 0.3t + 1.5) + {e^\omega } + {e^{ - \omega }}} \right)}^2}}},} \hfill \cr } where Q = (0.013 – 0.6i)x + 0.8iy – iz + 0.2t + 1.5 and ω = 0.4y + 0.3z + 0.4t + 1.5. Its propagation velocity along the xy-plane is shown in Fig. 10. Lump-breather interactions are elastic because each wave component retains its original velocity, amplitude, and shape; only the phase or position changes before and after the collision. Therefore, no energy transfer, energy dissipation, or change in waveform is observed within the asymptotic limits.

Physical interpretation of the findings

This section presents a comprehensive visual representation of the results from the KDKK and BK equations using the free parameters listed in Table 2, with an emphasis on the absolute components. To determine the amplitude and nature of the fluctuations in the solutions, we examine their absolute components to see how they vary along the absolute axis. This also helps us understand the overall stability and robustness of the solutions.

Table 2

Visual analysis of the KDKK and BK equations for the free parameters.

Fig. No.	Free parameters with intervals
Fig. 1	κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –2, κ₅ = –1, κ₆ = –2, l₁ = 1, m₂ = 1, n₂ = 1, w₁ = 1, δ₁ = δ₂ = 1, η = 1, z = 1 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 2	κ₁ = 2, κ₂ = 2, κ₃ = 1, κ₄ = –2, κ₅ = –1, κ₆ = 1, l₁ = l₂ = 1, n₁ = 1, w₁ = w₂ = 1, δ₁ = –1, δ₂ = 1, η = 1, t = 0 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 3	κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –2, κ₅ = κ₆ = 1, l₃ = 1, n₁ = n₂ = n₃ = 1, w₂ = 1, δ₁ = 1, δ₂ = 1, η = 1, z = 1, t = 0 for – 15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 4	κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –1, κ₅ = –1, κ₆ = 1, l₁ = l₃ = 1, l₂ = –1, n₁ = –3, n₂ = n₃ = 1, δ₁ = δ₂ = 1, η = 1, z = 5 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 5	κ₁ = 1, κ₂ = –1, κ₃ = 1, κ₄ = –3, κ₅ = κ6 = 1, m₂ = m₃ = 1, n₃ = 1, w₂ = 1, δ₁ = δ₂ = 1, η = 1 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 6	κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –1, κ₅ = 1, κ₆ = 1, m₁ = i, m₂ = m₃ = 1, n₁ = n₃ = 1, w₂ = 1, δ₁ = δ₂ = 1, $η = - \frac{1}{2}$ \eta = - {1 \over 2}, z = 3 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
Fig. 7	κ₁ = 1, κ₂ = 1, κ₃ = 1, κ₄ = –2, κ₅ = –1, κ₆ = 1, m₂ = m₃ = 1, n₁ = i, n₃ = –1, w₂ = 1, δ₁ = δ₂ = 1, η = 1, z = 1 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
Fig. 8	κ₁ = 1, κ₂ = 0.5, κ₃ = 1, κ₄ = –4.5, κ₅ = –1, κ₆ = 1, m₃ = –0.2, m₄ = 0.4, n₁ = 0.3, n₂ = –i, w₁ = 0.1, w₂ = 0.1, w3 = 0.2, w4 = 0.5, δ₁ = δ₂ = 1.5, η = –0.6, z = 0 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
Fig. 9	κ₁ = 2, κ₂ = 0.2, κ₃ = 1, κ₄ = –2, κ₅ = –1, κ₆ = 1, m₃ = –1, m₄ = 0.5, w₁ = 0.8, w₂ = 0.1, w₃ = 0.6, w₄ = 0.3, δ₁ = δ₂ = 1.5, η = –0.3, z = 0 for –40 ≤ x ≤ 20, –40 ≤ y ≤ 20
Fig. 10	κ₁ = 0.2, κ₂ = 0.4, κ₃ = 0.5, κ₄ = –3, κ₅ = –1, κ₆ = 1, m₃ = –0.2, m₄ = 0.4, n₁ = 0.3, n₂ = –i, w₁ = 0.1, w₂ = 0.2, w₃ = 0.3, w₄ = 0.4, δ₁ = δ₂ = δ₃ = δ₄ = 1.5, η = –0.6, z = 0 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30

In Fig. 1, we have illustrated the lump wave solution, where it propagates to the left along the x-axis in (a) and to the right in (b) and (c). The three-dimensional graph of the solution to equation (15) shows a peak in absolute value, and a contour plot displays a flower-like pattern that helps to understand the nature of the solution. Fig. 2 illustrates the nature of the lump wave solutions for different values of z in equation (18). The solutions are shifting to the right along the x-axis, shown as a 3D profile with a density plot and a flower-like contour plot. Fig. 3 illustrates the collision of the solitary and one-lump wave solutions of equation (22), where we have generated 3D profiles and density graphs using different parameters. Using time variation, we analyzed the collision of the solitary and one-lump wave solutions of equation (25) and displayed a 3D profile with density plots and 2D wave propagation along the x-axis, as shown in Fig. 4.

Based on different choices of free parameters and range space, Figs. 5, 6, and 7 illustrate the collision between the lump and periodic wave solutions of equations (29), (32), and (35), respectively. Fig. 5 presents the absolute parts corresponding to different soliton behaviors in the xy, xz, and xt planes. Additionally, Figs. 6 and 7 illustrate the behavior of the absolute value of the soliton with respect to time. The contour plots provide a detailed overview, which helps in understanding the nature of the solution and how the function behaves in the xy-plane. By selecting different values of y, the corresponding two-dimensional wave propagation along the x-axis is shown. In Figs. 8 and 9, we have illustrated the collisions between the lump, periodic, and solitary wave solutions of equations (39) and (42), respectively. Fig. 10 illustrates the collision of the breather and lump wave solutions of equation (46). Using time variation, the absolute solutions change their positions, which are represented as a 3D profile with a density plot and a 2D wave profile.

Analysis of bifurcations and equilibrium states

This section considers the following wave transformations, with an emphasis on the equilibrium and bifurcation analysis of the KDKK and BK equations (1): 47 $u = W (ξ), ξ = σ x + τ y + μ z - ρ t,$ u = W(\xi ),\quad \xi = \sigma x + \tau y + \mu z - \rho t, where σ, τ, μ, and ρ are free parameters. Inserting equation (47) into equation (1), which leads to the ordinary differential equation 48 $\frac{d^{2}}{d ξ^{2}} W (ξ) + (\frac{σ κ_{4} + τ κ_{5} + μ κ_{6} - ρ κ_{1}}{(τ κ_{3} + μ κ_{2}) σ^{2}}) W (ξ) + (\frac{3 σ μ κ_{3} + 3 τ^{2} κ_{3} + 6 κ_{2} μ τ}{2 (τ κ_{3} + μ κ_{2}) σ^{2} τ}) (^{W (ξ)) 2} = 0.$ {{{d^2}} \over {d{\xi ^2}}}W(\xi ) + \left( {{{\sigma {\kappa _4} + \tau {\kappa _5} + \mu {\kappa _6} - \rho {\kappa _1}} \over {\left( {\tau {\kappa _3} + \mu {\kappa _2}} \right){\sigma ^2}}}} \right)W(\xi ) + \left( {{{3\sigma \mu {\kappa _3} + 3{\tau ^2}{\kappa _3} + 6{\kappa _2}\mu \tau } \over {2\left( {\tau {\kappa _3} + \mu {\kappa _2}} \right){\sigma ^2}\tau }}} \right){(W(\xi ))^2} = 0.

Now, from equation (48), we get 49 $\frac{d^{2}}{d ξ^{2}} W (ξ) + χ_{1} W (ξ) + χ_{2} (^{W (ξ)) 2} = 0,$ {{{d^2}} \over {d{\xi ^2}}}W(\xi ) + {\chi _1}W(\xi ) + {\chi _2}{(W(\xi ))^2} = 0, where $χ_{1} = \frac{σ κ_{4} + τ κ_{5} + μ κ_{6} - ρ κ_{1}}{(τ κ_{3} + μ κ_{2}) σ^{2}}$ {\chi _1} = {{\sigma {\kappa _4} + \tau {\kappa _5} + \mu {\kappa _6} - \rho {\kappa _1}} \over {\left( {\tau {\kappa _3} + \mu {\kappa _2}} \right){\sigma ^2}}} and $χ_{2} = \frac{3 σ μ κ_{3} + 3 τ^{2} κ_{3} + 6 κ_{2} μ τ}{2 (τ κ_{3} + μ κ_{2}) σ^{2} τ}$ {\chi _2} = {{3\sigma \mu {\kappa _3} + 3{\tau ^2}{\kappa _3} + 6{\kappa _2}\mu \tau } \over {2\left( {\tau {\kappa _3} + \mu {\kappa _2}} \right){\sigma ^2}\tau }}.

To obtain the two-dimensional dynamic structure, rewrite equation (49) in the form: 50 $\begin{array}{l} \frac{d}{d ξ} W (ξ) = Y (ξ), \\ \frac{d}{d ξ} Y (ξ) = - χ_{1} W (ξ) - χ_{2} {(W (ξ))}^{2} . \end{array}$ \matrix{ {{d \over {d\xi }}W(\xi ) = Y(\xi ),} \hfill \cr {{d \over {d\xi }}Y(\xi ) = - {\chi _1}W(\xi ) - {\chi _2}{{(W(\xi ))}^2}.} \hfill \cr }

In this case, the Hamiltonian function H = H(W, Y) associated with the solution of the dynamic system (50), we can write 51 $H = \frac{1}{2} Y^{2} + \frac{χ_{1}}{2} W^{2} + \frac{χ_{2}}{3} W^{3} .$ H = {1 \over 2}{Y^2} + {{{\chi _1}} \over 2}{W^2} + {{{\chi _2}} \over 3}{W^3}.

The total energy of a dynamical system is denoted by H, which is the sum of the potential and kinetic energies of the particles, and it depends on their position and momentum. The paths along which the total energy remains constant are shown visually. Equation (51) satisfies $\frac{d W}{d ξ} = \frac{\partial H}{\partial Y}$ {{dW} \over {d\xi }} = {{\partial H} \over {\partial Y}} and $\frac{d Y}{d ξ} = - \frac{\partial H}{\partial W}$ {{dY} \over {d\xi }} = - {{\partial H} \over {\partial W}}. For the dynamic system (50), if $\frac{d W}{d ξ} = 0$ {{dW} \over {d\xi }} = 0 and $\frac{d Y}{d ξ} = 0$ {{dY} \over {d\xi }} = 0, then the equilibrium points (EPs) are (0, 0) and $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right). Let Tr and D be the trace and determinant of the Jacobian matrix (JM). Then, we can write: 52 $J M = [\begin{matrix} 0 & 1 \\ - χ_{1} - 2 χ_{2} W & 0 \end{matrix}], T r = 0, D = χ_{1} + 2 χ_{2} W .$ JM = \left[ {\matrix{ 0 & 1 \cr { - {\chi _1} - 2{\chi _2}W} & 0 \cr } } \right],\quad Tr = 0,\quad D = {\chi _1} + 2{\chi _2}W.

From the theory of planner dynamical systems, we can write the critical points of system (50) as follows: if D < 0 (D > 0), then EP is a saddle point (center), and if D = 0, then EP is a zero point. The eigenvalues of the dynamic system are $λ = \pm \sqrt{- χ_{1} - 2 χ_{2} W}$ \lambda = \pm \sqrt { - {\chi _1} - 2{\chi _2}W} . We first study the EPs to investigate the phase portraits of the dynamical system (50). For (0,0), the eigenvalues are $λ = \pm \sqrt{- χ_{1}}$ \lambda = \pm \sqrt { - {\chi _1}} , whereas $λ = \pm \sqrt{χ_{1}}$ \lambda = \pm \sqrt {{\chi _1}} are eigenvalues for the EP $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right). It is easily seen that if χ₁ ≠ 0, then the system (50) has two EPs, (0,0) and $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right). Meanwhile, if χ₁ = 0, then the system (50) has only one EP (0,0). All the equilibrium points lie on the W-axis. The study of dynamical systems involves four distinct phenomena, namely:

Case I. χ₁ > 0, χ₂ < 0

In this case, the EPs (0,0) and $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right) are the center and saddle point, respectively. Fig. 11a shows the phase portrait graph of the dynamic system (50) with free parameters. Here, the two extended curves of a homoclinic orbit are shown in green. There are two types of orbits in this family: the first type, shown in blue, consists of various periodic trajectories that rotate clockwise around the center point (0,0). The second type is a family of orbits that have a hyperbolic shape, highlighted in orange. Additionally, the homoclinic orbits are highlighted in magenta, representing the only remaining members of this family of orbits. Fig. 12 shows the Hamiltonian function of the dynamic system (51) through 2D, 3D, and contour graphics.

Case II. χ₁ > 0, χ₂ > 0

In this case, the EPs (0,0) and $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right) are the center and saddle point, respectively. Fig. 11b shows the phase portrait graph of the dynamic system (50) using the free parameters. Fig. 13 shows a graphical representation of the Hamiltonian function of system (51), including 2D, 3D, and contour plots. The discussion is similar to the previous Case I, which applies to both Figs. 11b and 13.

Case III. χ₁ < 0, χ₂ < 0

In this case, the EPs (0,0) and $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right) are the saddle and center points, respectively. Fig. 11c shows the phase portrait graph of the dynamic system (50) with free parameters. Here, the two extended curves of a homoclinic orbit are shown in green. There are two types of orbits in this family: the first type, shown in blue, consists of various periodic trajectories that rotate clockwise around the center point $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right). The second type is a family of orbits that have a hyperbolic shape, highlighted in orange. Additionally, the homoclinic orbits are highlighted in magenta, which are the only remaining members of this family of orbits. Fig. 14 shows the Hamiltonian function of the dynamic system (51) through 2D, 3D, and contour graphics.

Case IV. χ₁ < 0, χ₂ > 0

In this case, the EPs (0,0) and $(- \frac{χ_{1}}{χ_{2}}, 0)$ \left( { - {{{\chi _1}} \over {{\chi _2}}},0} \right) are the saddle and center points. Fig. 11d shows the phase portrait graph of the dynamic system (50) using the free parameters. Fig. 15 shows a visual representation of the Hamiltonian function of system (51), including 2D, 3D, and contour plots. The discussion is similar to the previous Case III, which applies to both Figs. 11d and 15.

Analysis of sensitivity

In this section, the dynamics of the KDKK and BK equations (1) are examined to see how small changes in parameters or initial conditions affect the behavior of a dynamic system. We analyze the sensitivity of the dynamic system (50) by utilizing the Runge-Kutta technique. The parameters κ₁ = 0.3, κ₂ = 0.5, κ₃ = 0.8, κ₄ = 0.8, κ₅ = 0.5, κ₆ = 0.3, σ = 1.2, τ = 1.1, μ = –0.8, and ρ = –0.5 are chosen to solve this system. To investigate the sensitivity of the system, we also consider four different initial conditions: IC1 [W(0) = 0.12, Y(0) = 0]; IC2 [W(0) = 0, Y(0) = 0.12]; IC3 [W(0) = 0.22, Y(0) = 0]; and IC4 [W(0) = 0, Y(0) = 0.22].

Additionally, Fig. 16 demonstrates the consistency of the unperturbed system (50) and analyzes how the system behaves under different initial conditions. This figure illustrates the sensitivity analysis of W(ξ), Y(ξ), and W(ξ) Y(ξ) with respect toξ, based on four distinct initial conditions. Fig. 16 shows four colors: cyan, blue, leaf green, and orange. Here, the cyan curve represents IC1 (0.12, 0), the blue curve represents IC2 (0, 0.12), the leaf green curve represents IC3 (0.22, 0), and the orange curve represents IC4 (0, 0.22), respectively. Leaf green indicates a long height curve, while blue indicates a short height curve. The graphs highlight the complex structure and variations of the system, making it easier to show how the system has changed over time under different initial conditions. Even small changes in the initial conditions can considerably affect the system’s behavior. Therefore, we identify that a minimum sensitivity is present in the given equation.

Conclusion

In this work, we applied the Hirota bilinear technique to determine the bilinear form of a generalized (3+1)-dimensional KDKK and BK equation that identifies the physical phenomenon of the waveform. Using a computer program known as Maple software, we illustrate the characteristics of collisions between different waves. The dynamical behaviour of lump waves and their collisions with single-lump and solitary waves, lump and periodic waves, and lump and breather waves has been examined and described using various ansatz functions. In addition, we illustrate how the lump interacts with breather waves, periodic waves, and solitary waves to generate different types of dynamical patterns in the solutions. The nature of the solutions is displayed visually, including 2D, 3D, and contour graphics. Figs. 1 and 2 show the dynamic structure and features of the lump wave solutions, whereas Figs. 3-10 display all the collisions between various waves. Also, the bifurcation process and sensitivity analysis of dynamic structures are studied based on the theory of planar dynamical systems. The exact stable Hamiltonian system is derived using a well-known Galilean transformation. We also demonstrate the phase diagram while performing a qualitative analysis of the Hamiltonian function. Therefore, the obtained results will help in understanding and describing the complex physical structure and characteristics of wave profiles related to fluid mechanics, shallow water waves, nonlinear dynamics, and ocean engineering. The advanced method proposed here can be easily applied to other nonlinear higher-dimensional systems, opening up new possibilities for studying wave-structure interactions in various computational engineering, optical fibers, nonlinear sciences, and other advanced scientific domains. In the future, researchers can extend the current research to multi-dimensional nonlinear mathematical models that include numerous physical parameters, and also investigate turbulent wave systems that more closely resemble real-world fluid dynamics and nonlinear dynamics.

Declarations

Dynamics of solitons, multi-lumps, and their interactions of the Konopelchenko-Dubrovsky-Kaup-Kupershmidt and Bogoyavlensky-Konopelchenko model in (3+1)-dimensions using the modern advanced approach

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