The Formation and Propagation of Soliton Wave Profiles for the Shynaray-IIa Equation

	P	Q	R	F
1	m²	−(1 + m²)	1	sn, cd
2	−m²	2m² − 1	1 − m²	cn
3	−1	2 − m²	m² − 1	dn
4	1	−(1 + m²)	m²	ns, dc
5	1 − m²	2m² − 1	−m²	nc
6	m² − 1	2 − m²	−1	nd
7	1 − m²	2 − m²	1	sc
8	−m²(1 − m²)	2m² − 1	1	sd
9	1	2 − m²	1 − m²	cs
10	1	2m² − 1	-m²(1 − m²)	ds
11	$\frac{- 1}{4}$	$\frac{m^{2} + 1}{2}$	$\frac{- {(1 - m^{2})}^{2}}{4}$	mcn ∓ dn
12	$\frac{1}{4}$	$\frac{- 2 m^{2} + 1}{2}$	$\frac{1}{4}$	ns ∓ cs
13	$\frac{1 - m^{2}}{4}$	$\frac{m^{2} + 1}{2}$	$\frac{1 - m^{2}}{4}$	nc ∓ sc
14	$\frac{1}{4}$	$\frac{m^{2} - 2}{2}$	$\frac{m^{4}}{4}$	ns ∓ ns
15	$\frac{m^{2}}{4}$	$\frac{m^{2} - 2}{2}$	$\frac{m^{2}}{4}$	$sn \mp icn, \frac{sn}{\sqrt{1 - m^{2} sn}}$
16	$\frac{1}{4}$	$\frac{1 - 2 m^{2}}{2}$	$\frac{1}{4}$	$mcn \mp idn, \frac{sn}{1 \mp cn}$
17	$\frac{m^{2}}{4}$	$\frac{m^{2} - 2}{2}$	$\frac{1}{4}$	$\frac{sn}{1 \mp dn}$
18	$\frac{m^{2} - 1}{4}$	$\frac{m^{2} + 1}{2}$	$\frac{m^{2} - 1}{4}$	$\frac{dn}{1 \mp msn}$
19	$\frac{1 - m^{2}}{4}$	$\frac{m^{2} + 1}{2}$	$\frac{- m^{2} + 1}{4}$	$\frac{cn}{1 \mp sn}$
20	$\frac{{(1 - m^{2})}^{2}}{4}$	$\frac{m^{2} + 1}{2}$	$\frac{1}{4}$	$\frac{sn}{dn \mp cn}$
21	$\frac{m^{4}}{4}$	$\frac{m^{2} - 2}{2}$	$\frac{1}{4}$	$\frac{cn}{\sqrt{1 - m^{2} \mp dn}}$

Tab. 2.

Analysis of Jacobi elliptic functions in the limit of m → 0 and m → 1.

		m → 1	m → 0			m → 1	m → 0
1	snu	tanhu	sinu	7	dcu	1	secu
2	cnu	sechu	cosu	8	ncu	coshu	secu
3	dnu	sechu	1	9	scu	sinhu	tanu
4	cdu	1	cosu	10	nsu	cothu	cscu
5	sdu	sinhu	sinu	11	dsu	cschu	cscu
6	ndu	coshu	1	12	csu	cschu	cotu

3-D, contour visualization and 2-D propagation of q_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.9, c = 0.1

3-D, contour visualization and 2-D propagation of q_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.9, c = 2.5

3-D, contour visualization and 2-D propagation of ν_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.9, c = 0.1.

3-D, contour visualization and 2-D propagation of ν_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.9, c = 2.5

3-D, contour visualization and 2-D propagation of q_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.5, c = 0.1

3-D, contour visualization and 2-D propagation of q_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.5, c = 2.5

3-D, contour visualization and 2-D propagation of ν_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.5, c = 0.1

3-D, contour visualization and 2-D propagation of ν_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.5, c = 2.5

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DOI: https://doi.org/10.2478/ama-2025-0016 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 136 - 147

Submitted on: Aug 6, 2024

Accepted on: Sep 25, 2024

Published on: Mar 31, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

soliton theory,

integrable partial differential equation,

analytical approach,

propagation of solitary waves

Related subjects:

Engineering,

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Muhammad Ishfaq Khan, Waqas Ali Faridi, Muhammad Amin Sadiq Murad, Mujahid Iqbal, Ratbay Myrzakulov, Zhanar Umurzakhova, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 1 (March 2025)

The Formation and Propagation of Soliton Wave Profiles for the Shynaray-IIa Equation

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