Phase-plane dynamics and nonlinear traveling-wave patterns in a (3+1)-dimensional mKdV-ZK equation

Jiaye Lin; Yaling Lai; Xiyan Wu; Changlong Chen; Yucheng Chen; Junjie Li; Jiaye Lin; Yaling Lai; Xiyan Wu; Changlong Chen; Yucheng Chen; Junjie Li

doi:10.3934/math.20261902

AIMS Mathematics

2026, Volume 11, Issue 8: 27304-27326. doi: 10.3934/math.20261902

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Research article

Phase-plane dynamics and nonlinear traveling-wave patterns in a (3+1)-dimensional mKdV-ZK equation

1.
School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, China
2.
College of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China

Received: 01 July 2026 Revised: 07 August 2026 Accepted: 11 August 2026 Published: 28 August 2026
MSC : 35Q53, 35C07

Traveling waves of the (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation are studied, first in the unforced case and then under a weak co-moving periodic source. Without forcing, the profile equation reduces to a quartic Hamiltonian whose level sets carry periodic waves, homoclinic pulses, heteroclinic kinks, and unbounded orbits. The substitution $ \Phi = U^2 $ turns the first integral into a cubic polynomial and gives a discriminant-based classification of a restricted family of explicit profiles subject to the constraint $ \Phi \ge 0 $ and to a piecewise sign reconstruction of $ U $. This part recovers structures already reported in bifurcation studies of the same model and serves as the geometric baseline. The forced problem is the main concern. A weak periodic source turns the reduction into a Duffing-type system for which closed-form Melnikov functions are obtained on both the homoclinic and the heteroclinic separatrix. Both have simple zeros, so both separatrices split transversely at a small forcing amplitude. Their amplitudes differ where it matters: The heteroclinic one tends to a finite positive limit as $ \omega \to 0 $, the homoclinic one vanishes there and peaks at an intermediate frequency, and both decay exponentially at high frequency. Stroboscopic Poincaré sections and largest Lyapunov exponents computed at $ \varepsilon = 0.01, 0.05 $, and $ 0.10 $ agree with this picture, the section passing from thin invariant arcs to a scattered layer around the former figure-eight as the exponent rises from order $ 10^{-4} $ to about $ 0.12 $.
- traveling-wave solution,
- bifurcation analysis,
- Melnikov analysis,
- chaotic dynamics,
- Hamiltonian system
Citation: Jiaye Lin, Yaling Lai, Xiyan Wu, Changlong Chen, Yucheng Chen, Junjie Li. Phase-plane dynamics and nonlinear traveling-wave patterns in a (3+1)-dimensional mKdV-ZK equation[J]. AIMS Mathematics, 2026, 11(8): 27304-27326. doi: 10.3934/math.20261902

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Abstract

Traveling waves of the (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation are studied, first in the unforced case and then under a weak co-moving periodic source. Without forcing, the profile equation reduces to a quartic Hamiltonian whose level sets carry periodic waves, homoclinic pulses, heteroclinic kinks, and unbounded orbits. The substitution $ \Phi = U^2 $ turns the first integral into a cubic polynomial and gives a discriminant-based classification of a restricted family of explicit profiles subject to the constraint $ \Phi \ge 0 $ and to a piecewise sign reconstruction of $ U $. This part recovers structures already reported in bifurcation studies of the same model and serves as the geometric baseline. The forced problem is the main concern. A weak periodic source turns the reduction into a Duffing-type system for which closed-form Melnikov functions are obtained on both the homoclinic and the heteroclinic separatrix. Both have simple zeros, so both separatrices split transversely at a small forcing amplitude. Their amplitudes differ where it matters: The heteroclinic one tends to a finite positive limit as $ \omega \to 0 $, the homoclinic one vanishes there and peaks at an intermediate frequency, and both decay exponentially at high frequency. Stroboscopic Poincaré sections and largest Lyapunov exponents computed at $ \varepsilon = 0.01, 0.05 $, and $ 0.10 $ agree with this picture, the section passing from thin invariant arcs to a scattered layer around the former figure-eight as the exponent rises from order $ 10^{-4} $ to about $ 0.12 $.

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