On the rational sine-Gordon solution of the forced KdV equation

The soliton theory has been an interesting topics of research for more than two hundred years. The fundamentals of this theory go back to J. Scott Russell [1], D. Korteweg and G. De Vries [2] and since then a large amount of papers have been dedicated to the soliton theory (see e.g. [3–6] and references therein). In recent years, after the discovering of nano scales and quantum world there has been a renewed interest for such a theory. In particular, among the most significant contributions let us remind some interesting soliton models in quantum theory (see e.g. [7, 8]), nano-materials (see e.g. [9, 10]), diffusion (see e.g. [11]), engineering (see e.g. [12]), mechanics (see e.g. [13]), dispersion (see e.g. [14, 15]), medicine (see e.g. [16, 17]), chemistry (see e.g. [18, 19]) and biology (see e.g. [20]).

In the following, we will study the forced KdV equation with a forcing term also known as KdV with forcing [21] 1ut(x,t)+αu(x,t)ut(x,t)+βuxxx(x,t)=F(t),{u_t}(x,t) + \alpha u(x,t){u_t}(x,t) + \beta {u_{xxx}}(x,t) = F(t), where uxxx(x,t)=∂3u(x,t)∂x3{u_{xxx}}(x,t) = {{{\partial ^3}u(x,t)} \over {\partial {x^3}}}, ut(x,t)=∂u(x,t)∂t{u_t}(x,t) = {{\partial u(x,t)} \over {\partial t}}, α and β are real constants, and, F(t) is a given differentiable function representing an outer time dependent forcing source. In nature, the forcing function F(t) is due to the bottom topography of a fluid domain (such as a bump on the bottom of a two dimensional channel), or due to some external pressures on the boundary domain such as the wind on the ocean surface [22].

If we consider the following function [23] 2u(x,t)=v(x,t)+∫0tF(t)dt,u(x,t) = v(x,t) + \int_0^t F (t)dt, then, Eq.(1) may be rewritten as follows 3vt(x,t)+αv(x,t)vx(x,t)+βvxxx(x,t)+α[∫0tF(t)dt]vx(x,t)=0,{v_t}(x,t) + \alpha v(x,t){v_x}(x,t) + \beta {v_{xxx}}(x,t) + \alpha [\int_0^t F (t)dt]{v_x}(x,t) = 0, where α, β are arbitrary constants.

In the following, we propose an analytical solution of equation (3) based on the rational sine-Gordon expansion method (RSGEM). This method is newly developed in the following and up to our knowledge it has never been applied to KdV equations in advance. RSGEM differ from the sine-Gordon expansion method (SGEM) because it contains rational trigonometric functions in the test function of the solution. Moreover, it is based on two important properties of the nonlinear sine-Gordon equation.

The organization of this paper is as follows: in Section 2, some preliminary remarks of RSGEM are given. Section 3 deals with the application of the RSGEM to obtain the analytical solution of the forced KdV equation. Some concrete discussion and physical comments of the solutions are reported in Section 4. Section 5 concludes the paper by summarizing the main novelties of this paper and discussing about future prospectives.

In this part of the paper, the main properties of RSGEM are given. Let’s consider the sine-Gordon equation [24, 25] 4uxx−utt=m2sin(u),{u_{xx}} - {u_{tt}} = {m^2}\sin (u), where m is an arbitrary constant. By using u = u(x, t) = U(ξ), ξ = μ(x — ct), Eq.(4) can be rewritten as 5U″=m2μ2(1−c2)sin(U),U'' = {{{m^2}} \over {{\mu ^2}(1 - {c^2})}}\sin (U), where U=U(ξ),U″(ξ)=d2U(ξ)dξ2U = U(\xi ),U''(\xi ) = {{{d^2}U(\xi )} \over {d{\xi ^2}}} and c,μ are arbitrary constants. After some basic calculations, we have 6((U2)′)2=m2μ2(1−c2)sin2(U2)+k,{({({U \over 2})^\prime })^2} = {{{m^2}} \over {{\mu ^2}(1 - {c^2})}}{\sin ^2}({U \over 2}) + k, k being a nonzero real constant. By assuming k = 0, w=U2w = {U \over 2}, and a2=m2μ2(1−c2){a^2} = {{{m^2}} \over {{\mu ^2}(1 - {c^2})}}, equation (6) becomes 7w′=asin(w),w' = a\sin (w), where w = w(ξ). By solving this equation with a = 1 we have 8sin(w)=sin[w(ξ)]=limp=12peξp2e2ξ+1=sech(ξ),\sin (w) = \sin [w(\xi )] = li{m_{p = 1}}{{2p{e^\xi }} \over {{p^2}{e^{2\xi }} + 1}} = {\rm{sech}}(\xi ), 9cos(w)=cos[w(ξ)]=limp=12peξp2e2ξ+1=tanh(ξ).\cos (w) = \cos [w(\xi )] = li{m_{p = 1}}{{2p{e^\xi }} \over {{p^2}{e^{2\xi }} + 1}} = {\rm{tanh}}(\xi ).

In this section, different types of soliton solutions to the KdV equation (3) with a forcing term are extracted by using RSGEM. Let’s take 14v=V(η),η=kx+∫0tτ(t)dt,v = V(\eta ),\eta = kx + \int_0^t \tau (t)dt, where k is real constant. τ(t) will be determined later. Putting equation (14) into equation (3), we obtain 15τ(t)V′(η)+kαV(η)V′(η)+kαΨ(t)V′(η)+k3βV‴(η)=0,\tau (t)V'(\eta ) + k\alpha V(\eta )V'(\eta ) + k\alpha \Psi (t)V'(\eta ) + {k^3}\beta V'''(\eta ) = 0, where V′=dV(η)dtV' = {{dV(\eta )} \over {dt}} and V‴=d3V(η)dt3V''' = {{{d^3}V(\eta )} \over {d{t^3}}}. In equation (15), by using balance rule, we obtain N=M=1. So, the trial solution formula may be written as 16V(w)=A0+A1sin(w)+c1cos(w)B0+B1sin(w)+d1cos(w),V(w) = {{{A_0} + {A_1}\sin (w) + {c_1}\cos (w)} \over {{B_0} + {B_1}\sin (w) + {d_1}\cos (w)}}, where A₀ ≠ B₀, A₁ ≠ B₁, c₁ ≠ d₁ simultaneously, and also both B₁ and d₁ must be not taken as zero at the same time. If these are A₀ = B₀, A₁ = B₁, c₁ = d₁ in this case, the solution is trivial. Using equation (16) into (15), we obtain the system of an algebraic equation being with a different order. Solving this system via computational program, we obtain the following set of results.

Set 1. The first set of results is given by the following 17A0=B0c1d1,B1=B02−d12,β=α6k2(−c1d1+A1B02−d12),τ(t)=κα[∫0tF(t)dt+5c16d1+A16B02−d12].{A_0} = {{{B_0}{c_1}} \over {{d_1}}},{B_1} = \sqrt {B_0^2 - d_1^2},\beta = {\alpha \over {6{k^2}}}( - {{{c_1}} \over {{d_1}}} + {{{A_1}} \over {\sqrt {B_0^2 - d_1^2} }}),\tau (t) = \kappa \alpha [\int_0^t F (t)dt + {{5{c_1}} \over {6{d_1}}} + {{{A_1}} \over {6\sqrt {B_0^2 - d_1^2} }}]. From (12), (14) and (17), the hyperbolic travelling wave solution of equation (3) follows 18v1(x,t)=A1sech(kx+τ(t)t)+c1(B0d1+tanh(kx+τ(t)t))B0+B02−d12sech(kx+τ(t)t)+d1tanh(kx+τ(t)t),{v_1}(x,t) = {{{A_1}sech(kx + \tau (t)t) + {c_1}({{{B_0}} \over {{d_1}}} + \tanh (kx + \tau (t)t))} \over {{B_0} + \sqrt {B_0^2 - d_1^2} sech(kx + \tau (t)t) + {d_1}\tanh (kx + \tau (t)t)}}, where k ≠ 0, A₁ ≠ 0, c₁ ≠ 0, B₀ ≠ 0, d₁ ≠ 0 and τ(t) is an arbitrary function given by equation (14).

To observe the wave dynamics of equation (18) under the various properties of τ(t), we consider the trigonometric functions of the τ(t) such as sin(t), cos(t), tan(t) and cot(t). In Figure 1, the new and novel main wave propagations are plotted under the real data. These simulations are firstly presented to the literature.

Fig. 1

Parametric nature wave behaviors for equation (18) at (a) : τ(t) = sin(t), (b) : τ(t) = cos(t), (c) : τ(t) = tan(t) and (d) : τ(t) = cot(t).

Set 2. We may consider other coefficients as follows 19B1=ϖ,A1=((−5k3β+τ(t))A0−6k3βΨ(t)B0)ϖ(k3β+τ(t))B0,α=−(k3β+τ(t))B0k(A0+Ψ(t)B0),c1=A0d1B0,{B_1} = \varpi,{A_1} = {{(( - 5{k^3}\beta + \tau (t)){A_0} - 6{k^3}\beta \Psi (t){B_0})\varpi } \over {({k^3}\beta + \tau (t)){B_0}}},\alpha = - {{({k^3}\beta + \tau (t)){B_0}} \over {k({A_0} + \Psi (t){B_0})}},{c_1} = {{{A_0}{d_1}} \over {{B_0}}}, where ϖ=B02−d12,Ψ(t)=∫0tF(t)dt\varpi = \sqrt {B_0^2 - d_1^2},\Psi (t) = \int_0^t F (t)dt.. From (12), (14) and (19), the following other periodic solution of equation (3) is obtained 20v2(x,t)=A0+1B0k3β+τ(t)B0(sech(kx+τ(t)t)((−5k3βA0+τ(t)A0−6k3βΨ(t)B0)ϖ+A0d1B0tanh(kx+τ(t)t)B0+B02−d12sech(kx+τ(t)t)+d1tanh(kx+τ(t)t),{v_2}(x,t) = {{{A_0} + {1 \over {{B_0}{k^3}\beta + \tau (t){B_0}}}(sech(kx + \tau (t)t)(( - 5{k^3}\beta {A_0} + \tau (t){A_0} - 6{k^3}\beta \Psi (t){B_0})\varpi + {{{A_0}{d_1}} \over {{B_0}}}\tanh (kx + \tau (t)t)} \over {{B_0} + \sqrt {B_0^2 - d_1^2} sech(kx + \tau (t)t) + {d_1}\tanh (kx + \tau (t)t)}}, where k ≠ 0, A₀ ≠ 0, d₁ ≠ 0, B₀ ≠ 0, ϖ=B02−d12\varpi = \sqrt {B_0^2 - d_1^2} .

The dynamical distribution of equation (20) according to the various properties such as F(t) = sin(t), F(t) = cos(t), F(t) = tan(t) and F(t) = cot(t), respectively and also τ(t) = cos(t), are simulated in Figure 2. In Figure 2, we see the important properties such as singular and travelling wave propagations of the governing model. These behavior are used to symbolize the real distribution in hyperbolic domain.

Fig. 2

Parametric nature wave behaviors for equation (20) at (a) : F(t) = sin(t), (b) : F(t) = cos(t), (c) : F(t) = tan(t) and (d) : F(t) = cot(t), and also τ(t) = cos(t).

Set 3. If we consider following coefficients, 21B1=−ϖ,A1=((−5k3β−τ(t))A0+6k3βΨ(t)B0)ϖ(k3β+τ(t))B0,α=−(k3β+τ(t))B0k(A0+Ψ(t)B0),c1=A0d1B0,{B_1} = - \varpi,{A_1} = {{(( - 5{k^3}\beta - \tau (t)){A_0} + 6{k^3}\beta \Psi (t){B_0})\varpi } \over {({k^3}\beta + \tau (t)){B_0}}},\alpha = - {{({k^3}\beta + \tau (t)){B_0}} \over {k({A_0} + \Psi (t){B_0})}},{c_1} = {{{A_0}{d_1}} \over {{B_0}}}, where ϖ=B02−d12,Ψ(t)=∫0tF(t)dt\varpi = \sqrt {B_0^2 - d_1^2},\Psi (t) = \int_0^t F (t)dt, we obtain the next soliton solution. From (12), (14) and (21), the following other version of the periodic solution of equation (3) 22v3(x,t)=A0+1B0k3β+τ(t)B0sech(kx+τ(t)t)(5k3βA0−τ(t)A0+6k3βΨ(t)B0)ϖ+A0d1B0tanh(kx+τ(t)t)B0−B02−d12sech(kx+τ(t)t)+d1tanh(kx+τ(t)t),{v_3}(x,t) = {{{A_0} + {1 \over {{B_0}{k^3}\beta + \tau (t){B_0}}}sech(kx + \tau (t)t)(5{k^3}\beta {A_0} - \tau (t){A_0} + 6{k^3}\beta \Psi (t){B_0})\varpi + {{{A_0}{d_1}} \over {{B_0}}}\tanh (kx + \tau (t)t)} \over {{B_0} - \sqrt {B_0^2 - d_1^2} sech(kx + \tau (t)t) + {d_1}\tanh (kx + \tau (t)t)}}, where k ≠ 0, A₀ ≠ 0, β ≠ 0, d₁ ≠ 0, ϖ=B02−d12\varpi = \sqrt {B_0^2 - d_1^2} .

The propagation of equation (22) by considering F(t) = sin(t) cos(t) tan(t) and τ(t) = cot(t) are plotted in Figure 3 and and Figure 4. In these figures, one can see that the equation (22) has single periodic propagations in governing model.

Fig. 3

Parametric nature wave behaviors for equation (22) at F(t) = sin(t) cos(t) tan(t) is (a)3D, and (b) Contour, respectively, and also τ = cos(t).

Fig. 4

2D wave behavior for equation (22) at F(t) = sin(t) cos(t) tan(t) with 2D, and t = 8.5, and also τ(t) = cos(t), τ(t) = sin(t), respectively.

Set 4. For the proceeding to the next solution, we may take and 23B0=B12+d12,A1=3k4β2B12(B12+d12)2αB12+αd12+B1(3k2βα+A0B12+d12),c1=A0d1B12+d12,{B_0} = \sqrt {B_1^2 + d_1^2},{A_1} = {{3\sqrt {{k^4}{\beta ^2}B_1^2{{(B_1^2 + d_1^2)}^2}} } \over {\alpha B_1^2 + \alpha d_1^2}} + {B_1}({{3{k^2}\beta } \over \alpha } + {{{A_0}} \over {\sqrt {B_1^2 + d_1^2} }}),{c_1} = {{{A_0}{d_1}} \over {\sqrt {B_1^2 + d_1^2} }}, and 24F(t)=−ddt[τ(t)kα−A0B12+d12−3B1k4β2B12(B12+d12)2+k2β(B12+d12)(2B12+5d12)5α(B12+d12)2].F(t) = - {d \over {dt}}[{{\tau (t)} \over {k\alpha }} - {{{A_0}} \over {\sqrt {B_1^2 + d_1^2} }} - {{3{B_1}\sqrt {{k^4}{\beta ^2}B_1^2{{(B_1^2 + d_1^2)}^2}} + {k^2}\beta (B_1^2 + d_1^2)(2B_1^2 + 5d_1^2)} \over {5\alpha {{(B_1^2 + d_1^2)}^2}}}]. From (12), (14), (23) and (24), the following smooth travelling wave solution of equation (3) is obtained as below 25v4(x,t)=A0B12+d12+3sech(kx+τ(t)t)(k2βB1+k4β2B12(B12+d12)2B12+d12)αB1sech(kx+τ(t)t)+B12+d12+d1tanh(kx+τ(t)t),{v_4}(x,t) = {{{A_0}} \over {\sqrt {B_1^2 + d_1^2} }} + {{3sech(kx + \tau (t)t)({k^2}\beta {B_1} + {{\sqrt {{k^4}{\beta ^2}B_1^2{{(B_1^2 + d_1^2)}^2}} } \over {B_1^2 + d_1^2}})} \over {\alpha {B_1}sech(kx + \tau (t)t) + \sqrt {B_1^2 + d_1^2} + {d_1}\tanh (kx + \tau (t)t)}}, where k ≠ 0, A₀ ≠ 0, d₁ ≠ 0, B₁ ≠ 0.

The equation (25) contains only three hyperbolic function solution terms. This solution may not be obtained by getting other values of the parameters in other cases. So, by considering τ(t) = cos(t), we plot a realistic wave simulation and also observe the wave propagation of equation (25) in Figure 5.

Fig. 5

The wave way behavior for equation (25) at τ(t) = cos(t) with a) 3D and b) Contour.

In this part of the paper, we present the physical properties of the solution obtained in this paper. The parametric nature wave behaviors for equation (18) are plotted in Figure 1. This figure symbolizes the effects of the forcing term in equation (3). This wave simulation is newly presented to literature. In Figure 2, we observe the effect of F(t) under the different trigonometric functions. All these functions of F(t) shows the characteristic property of the equation (3). In Figure 3, it helps to investigate the periodic behaviour of the equation (3) model with forcing term. In Figure 4, we observe the singular wave behavior of equation (3). In Figure 5, it helps to understand the stable wave character of the power forcing term. All these graphical simulations are firstly plotted. Moreover, they introduce the expected wave behaviors. Thus, these figures help to understand the significance of forcing terms. Hence, it is estimated that these results may be used to estimated the future direction of the problem.

In this paper, first of all, we presented the recently developed RSGEM. RSGEM is based on the main properties of the sine-Gordon equation. With the help of RSGEM, the KdV equation with a forcing term was investigated in detailed manner. We obtained various travelling wave solutions of the model such as hyperbolic, and rational function solutions. We also observed that all these solutions verified the KdV equation with a forcing term (3). Then, we plotted these solutions under the considering various values of the parameters by using computational program. These figures were plotted in 3D, 2D and density (contour) frame. In the next section, we presented the physical meanings of the figures. We observed the importance of the forcing term of the model. The current investigation can help to illustrate the model associated with parameters representing water wave surface. To the best of our knowledge, these solutions of KdV model with forcing term have been firstly literature.