Exploring Jacobi Elliptic and Periodic Solitary Wave Solutions for the Family of 3-D WBBM Equations Through the Generalized Approach

NO.	m₂	m₁	m₀	N
1	k²	–(1 + k²)	1	sn
2	k²	–(1 + k²)	1	cd
3	–k²	2k² – 1	1 – k²	cn
4	–1	2 – k²	k² – 1	dn
5	1	–(1 + k²)	k²	ns
6	1	–(1 + k²)	k²	dc
7	1 – k²	2k² – 1	–k²	nc
8	k² – 1	2 – k²	–1	nd
9	1 – k²	2 – k²	1	sc
10	–k²(1 – k²)	2k² – 1	1	sd
11	1	2 – k²	1 – k²	cs
12	1	2k² – 1	-k²(1 – k²)	ds
13	$\frac{- 1}{4}$	$\frac{k^{2} + 1}{2}$	$\frac{- {(1 - k^{2})}^{2}}{4}$	kcn ∓ dn
14	$\frac{1}{4}$	$\frac{- 2 k^{2} + 1}{2}$	$\frac{1}{4}$	ns ∓ cs
15	$\frac{1 - k^{2}}{4}$	$\frac{k^{2} + 1}{2}$	$\frac{1 - k^{2}}{4}$	nc ∓ sc
16	$\frac{1}{4}$	$\frac{k^{2} - 2}{2}$	$\frac{k^{4}}{4}$	ns ∓ ds
17	$\frac{k^{2}}{4}$	$\frac{k^{2} - 2}{2}$	$\frac{k^{2}}{4}$	sc ∓ icn
18	$\frac{k^{2}}{4}$	$\frac{k^{2} - 2}{2}$	$\frac{k^{2}}{4}$	$\frac{s n}{\sqrt{1 - k^{2} s n \mp c n}}$
19	$\frac{k^{2}}{4}$	$\frac{k^{2} - 2}{2}$	$\frac{k^{2}}{4}$	kcn ∓ idn
20	$\frac{1}{4}$	$\frac{1 - 2 k^{2}}{2}$	$\frac{1}{4}$	$\frac{s n}{1 \mp c n}$
21	$\frac{k^{2}}{4}$	$\frac{k^{2} - 2}{2}$	$\frac{1}{4}$	$\frac{s n}{1 \mp d n}$
22	$\frac{k^{2} - 1}{4}$	$\frac{k^{2} + 1}{2}$	$\frac{k^{2} - 1}{4}$	$\frac{d n}{1 \mp k s n}$
23	$\frac{1 - k^{2}}{4}$	$\frac{k^{2} + 1}{2}$	$\frac{- k^{2} + 1}{4}$	$\frac{c n}{1 \mp s n}$
24	$\frac{{(1 - k^{2})}^{2}}{4}$	$\frac{k^{2} + 1}{2}$	$\frac{1}{4}$	$\frac{s n}{d n \mp c n}$
25	$\frac{k^{4}}{4}$	$\frac{k^{2} - 2}{2}$	$\frac{1}{4}$	$\frac{c n}{\sqrt{1 - k^{2} \mp d n}}$

Tab. 2.

The Jacobi elliptic function exhibit specific limiting behaviour as k → 0 and k → 1

No.	Function	k → 1	k → 0
1	sn(v)	tanh(v)	sin(v)
2	cn(v)	sech(v)	cos(v)
3	dn(v)	sech(v)	1
4	cd(v)	1	cos(v)
5	sd(v)	sinh(v)	sin(v)
6	nd(v)	cosh(v)	1
7	dc(v)	1	sec(v)
8	nc(v)	cosh(v)	sec(v)
9	sc(v)	sinh(v)	tan(v)
10	ns(v)	coth(v)	csc(v)
11	ds(v)	csch(v)	csc(v)
12	cs(v)	csch(v)	ct(v)

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,1(*x, y, z*) are presented for the parameter values k = 0.5, r = 1, p = 2, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of v1,3(x, y, z) are presented for the parameter values k = 0.5, r = 0.5, p = 2, q = 2, and y = z = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,4(*x, y, z*) are presen-ted for the parameter valuesk = 0.5, r = 0.5, p = 2, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,5(*x, y, z*) are presen-ted for the parameter values k = 0.5, r = 1, p = 2, q = 2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,11(*x, y, z*) are presen-ted for the parameter values k = 0.5, r = 1.5, p = –2, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,26(*x, y, z*) are presented for the parameter values r = 2.5, p = 2, q = 𠀓2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,27(*x, y, z*) are pre-sented for the parameter values r = 3, p = 1.5, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,29(*x, y, z*) are presen-ted for the parameter values r = 1.8, p = 1.5, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,40(*x, y, z*) are presented for the parameter values r = 1.8, p = 1.5, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,41(*x, y, z*) are pre-sented for the parameter values r = 1.8, p = 1.5, q = –2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_1,43(*x, y, z*) are pre-sented for the parameter values r = 1.8, p = 1.5, q = 2, and *y = z* = 1

The 3-D visualization, contour plot, and 2-D gra-phical representation of V_2,1(*x, y, z*) are presented for the parameter values k = 0.5, r = –1, p = 2, q = 2, y = 1, and z = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_2,3(*x, y, z*) are pre-sented for the parameter values k = 0.5, r = –1, p = 2, q = 1.5, y = 1, and z = 1

The 3-D visualization, contour plot, and 2-D graphical representation of V_3,1(*x, y, z*) are pre-sented for the parameter values k = 0.5, r = 1, p = 2, q = –2, y = 1, and z = 1

The 3-D visualization, contour plot and 2-D graphical representation of V_3,3(*x, y, z*) are presented for the parameter values k = 0.6, r = 0.5, p = 2, q = 2, and *y = z* = 1

References

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DOI: https://doi.org/10.2478/ama-2025-0048 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 406 - 425

Submitted on: Feb 24, 2025

Accepted on: Jul 15, 2025

Published on: Sep 5, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

Soliton wave solutions,

family of 3-D WBBM Equations,

Jacobi Elliptic Function Expansion technique,

Nonlinear Evaluation Equations

Related subjects:

Engineering,

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Muhammad Ishfaq KHAN, Kalim ULLAH, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 3 (September 2025)

Exploring Jacobi Elliptic and Periodic Solitary Wave Solutions for the Family of 3-D WBBM Equations Through the Generalized Approach

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