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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.11287 |
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Table of Contents:
- When the phase shift of X-shaped solutions before and after interaction is finite but approaches infinity, the vertices of the two V-shaped structures become separated due to the phase shift and are connected by a localized structure, which is referred to as the stem structure. This special type of elastic collision is known as a quasi-resonant collision. This study investigates quasi-resonant solutions and the associated localized stem structures in the context of the KPII and KPI equations. For the KPII equation, we classify quasi-resonant 2-solitons into weakly and strongly types, depending on whether the phase approaches \(-\infty\) or \(+\infty\). We analyze their asymptotic forms to detail the trajectories, amplitudes, velocities, and lengths of their stem structures. These results of quasi-resonant 2-solitons are used to to provide analytical descriptions of interesting patterns of the water waves observed on shallow water surface. Similarly, for the KPI equation, we construct quasi-resonant breather-soliton solutions and classify them into weakly and strongly types, based on the behavior of their internal parameters. We compare the similarities and differences between the stem structures in the quasi-resonant soliton and the quasi-resonant breather-soliton. Additionally, we provide a comprehensive and rigorous analysis of their asymptotic forms and stem structures. Our results indicate that the resonant solution, i.e. resonant breather-soliton of the KPI and soliton for the KPII, represents the limiting case of the quasi-resonant solution as phase approaches \(\infty\).