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

KHAN, Muhammad Ishfaq; ULLAH, Kalim

doi:10.2478/ama-2025-0048

Figures & Tables

The 3-D visualization, contour plot, and 2-D graphical representation of V1,1(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 V1,4(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 V1,5(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 V1,11(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 V1,26(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 V1,27(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 V1,29(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 V1,40(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 V1,41(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 V1,43(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 V2,1(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 V2,3(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 V3,1(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 V3,3(x, y, z) are presented for the parameter values k = 0.6, r = 0.5, p = 2, q = 2, and y = z = 1

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 following table summarizes all the potential solutions to equation (2_3) for the specific m2, m1, and m0 values that have been provided

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	−14{{ - 1} \over 4}	k2+12{{{k^2} + 1} \over 2}	−(1−k2)24{{ - {{\left( {1 - {k^2}} \right)}^2}} \over 4}	kcn ∓ dn
14	14{1 \over 4}	−2k2+12{{ - 2{k^2} + 1} \over 2}	14{1 \over 4}	ns ∓ cs
15	1−k24{{1 - {k^2}} \over 4}	k2+12{{{k^2} + 1} \over 2}	1−k24{{1 - {k^2}} \over 4}	nc ∓ sc
16	14{1 \over 4}	k2−22{{{k^2} - 2} \over 2}	k44{{{k^4}} \over 4}	ns ∓ ds
17	k24{{{k^2}} \over 4}	k2−22{{{k^2} - 2} \over 2}	k24{{{k^2}} \over 4}	sc ∓ icn
18	k24{{{k^2}} \over 4}	k2−22{{{k^2} - 2} \over 2}	k24{{{k^2}} \over 4}	sn1−k2sn∓cn{{sn} \over {\sqrt {1 - {k^2}sn \mp cn} }}
19	k24{{{k^2}} \over 4}	k2−22{{{k^2} - 2} \over 2}	k24{{{k^2}} \over 4}	kcn ∓ idn
20	14{1 \over 4}	1−2k22{{1 - 2{k^2}} \over 2}	14{1 \over 4}	sn1∓cn{{sn} \over {1 \mp cn}}
21	k24{{{k^2}} \over 4}	k2−22{{{k^2} - 2} \over 2}	14{1 \over 4}	sn1∓dn{{sn} \over {1 \mp dn}}
22	k2−14{{{k^2} - 1} \over 4}	k2+12{{{k^2} + 1} \over 2}	k2−14{{{k^2} - 1} \over 4}	dn1∓ksn{{dn} \over {1 \mp ksn}}
23	1−k24{{1 - {k^2}} \over 4}	k2+12{{{k^2} + 1} \over 2}	−k2+14{{ - {k^2} + 1} \over 4}	cn1∓sn{{cn} \over {1 \mp sn}}
24	(1−k2)24{{{{\left( {1 - {k^2}} \right)}^2}} \over 4}	k2+12{{{k^2} + 1} \over 2}	14{1 \over 4}	sndn∓cn{{sn} \over {dn \mp cn}}
25	k44{{{k^4}} \over 4}	k2−22{{{k^2} - 2} \over 2}	14{1 \over 4}	cn1−k2∓dn{{cn} \over {\sqrt {1 - {k^2} \mp dn} }}

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

Figures & Tables

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

The following table summarizes all the potential solutions to equation (2_3) for the specific m2, m1, and m0 values that have been provided

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