Dynamical bifurcations of nontrivial traveling waves in the Zakharov-Ito equation: a Hamiltonian system approach with parameter dependence

Fan Wu; Jianhong Zhuang; Yaqing Liu; Fan Wu; Jianhong Zhuang; Yaqing Liu

doi:10.3934/math.2026270

AIMS Mathematics

2026, Volume 11, Issue 3: 6525-6559. doi: 10.3934/math.2026270

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Dynamical bifurcations of nontrivial traveling waves in the Zakharov-Ito equation: a Hamiltonian system approach with parameter dependence

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

Received: 25 December 2025 Revised: 08 February 2026 Accepted: 28 February 2026 Published: 13 March 2026
MSC : 34C37, 34C60, 35B32, 35C07

This paper proposes a Hamiltonian framework to systematically classify nontrivial traveling wave classical solutions for the Zakharov-Ito (ZI) equation defined on $ \mathbb R $, revealing their parameter dependence and dynamic bifurcation features. The rigorous analysis proves that only two types of smooth traveling wave solutions exist for the ZI equation in the real domain: periodic solutions and unimodal solutions (which include solitary waves), with both left-going and right-going waves being identified. These findings differ from a previous result that asserted the existence of compactons and kink-type solutions. This work not only establishes complete existence theorems for ZI traveling waves but also highlights a subtlety in certain coordinate transformation techniques which may inadvertently introduce solutions that do not correspond to genuine traveling waves of the original ZI system. This work establishes the foundation for future investigations into the stability of traveling wave solutions.
- Zakharov-Ito equation,
- traveling wave solutions,
- periodic solution,
- soliton solution,
- Hamiltonian system
Citation: Fan Wu, Jianhong Zhuang, Yaqing Liu. Dynamical bifurcations of nontrivial traveling waves in the Zakharov-Ito equation: a Hamiltonian system approach with parameter dependence[J]. AIMS Mathematics, 2026, 11(3): 6525-6559. doi: 10.3934/math.2026270

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Abstract

This paper proposes a Hamiltonian framework to systematically classify nontrivial traveling wave classical solutions for the Zakharov-Ito (ZI) equation defined on $ \mathbb R $, revealing their parameter dependence and dynamic bifurcation features. The rigorous analysis proves that only two types of smooth traveling wave solutions exist for the ZI equation in the real domain: periodic solutions and unimodal solutions (which include solitary waves), with both left-going and right-going waves being identified. These findings differ from a previous result that asserted the existence of compactons and kink-type solutions. This work not only establishes complete existence theorems for ZI traveling waves but also highlights a subtlety in certain coordinate transformation techniques which may inadvertently introduce solutions that do not correspond to genuine traveling waves of the original ZI system. This work establishes the foundation for future investigations into the stability of traveling wave solutions.

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