Localized stem structures in quasi-resonant solutions of the Kadomtsev-Petviashvili equation

TL;DR

This paper analyzes quasi-resonant collisions in KPI and KPII, focusing on localized stem structures that connect two V-shaped wavefronts during elastic interactions. Using Hirota bilinear tau functions, it derives precise asymptotic forms for the stem and arm structures in both KPII (two-soliton) and KPI (breather-soliton) settings, classifying weak and strong quasi-resonances via $a_{12}$ and $(\alpha_1^2+\beta_1^2)$. It demonstrates that resonant (Y-shaped) solutions arise as the limit $\epsilon\to0$ and shows how the stem length diverges in this limit, effectively bridging X- and Y-shaped behaviors. The work applies the KPII results to Venice Beach water-wave observations and presents analogous quasi-resonant breather-soliton structures for KPI, together providing a unified analytical framework for stem structures in quasi-resonant KPI-type equations. Overall, the results offer quantitative predictions for stem trajectories, endpoints, and lengths, enabling direct comparison with experimental and oceanic wave patterns.

Abstract

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. This special type of elastic collision is known as a quasi-resonant collision, and the localized structure is referred to as the stem structure. 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 parameter \(a_{12} \approx 0\) 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 Venice Beach. Similarly, for the KPI equation, we construct quasi-resonant breather-soliton solutions and classify them into weakly and strongly types, based on whether the parameters \(α_1^2 + β_1^2 \approx 0\) or \(+\infty\) (equivalent to \(a_{13} \approx 0\) or \(+\infty\)). 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 \(ε\to 0\).

Figures (9)

• Figure 1: The weakly quasi-resonant 2-soliton $u_{qw}^{[2]}$ with $k_1=\frac{5}{3},\,k_2=1,\,p_1=1,\,p_2=\frac{ k_2(k_1^{2}-k_1k_2+ p_1)}{k_1}-\epsilon,\,\epsilon=10^{-7} ,\,t=0$. (a) 3D map; (b) The density plot and trajectories; (c) The section-cross curve $u|_{l_{1-2}}$.
• Figure 2: Parameters: $k_1=\frac{5}{3},\,k_2=1,\,k_3=1,\,p_1=1,\,p_2=\frac{ k_2(k_1^{2}-k_1k_2+ p_1)}{k_1}-\epsilon ,\,t=0$. (a) Graphs of $L_{A_1B_1}$ and $|\Delta_{12}|$ as the function of $\epsilon$; (b) The trajectories of $u_{qw}^{[2]}$ with different $\epsilon$; (c) The section curves $u|_{l_{1-2}}$ with different $\epsilon$.
• Figure 3: The strongly quasi-resonant 2-soliton with $k_1=\frac{5}{3},\,k_2=1,\,p_1=1,\,p_2=-\frac{k_2(k_1^{2}+k_1k_2-p_1)}{k_1}-\epsilon,\,\epsilon=10^{-7} ,\,t=0$. (a) 3D map; (b) The density plot and trajectories; (c) The section-cross curve $u|_{l_{1+2}}$
• Figure 4: Parameters: $k_1=\frac{5}{3},\,k_2=1,\,p_1=1,\,p_2=-\frac{k_2(k_1^{2}+k_1k_2-p_1)}{k_1}-\epsilon ,\,t=0$. (a) Graphs of $L_{A_2B_2}$ and $|\Delta_{12}|$ as the function of $\epsilon$; (b) The trajectories of $u_{qs}^{[2]}$ with different $\epsilon$; (c) The section curves $u|_{l_{1+2}}$ with different $\epsilon$.
• Figure 5: Comparison between plots(Fig.(a,b)) of quasi-resonant 2 soliton and photograph (c) for patterns of ocean wave. The photograph originates from Figs. 3 (b) of Ref. kppre2012. Here, I and II denote soliton-1 and soliton-2, while $-$ and $+$ denote before and after collision, respectively. Parameters: $k_1=k_2=\frac{1}{2},\,p_1=-\frac{1}{8}-10^{-8},\,p_2=\frac{3}{8},\,t=0$.

Theorems & Definitions (9)

• Remark 1
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• Remark 5
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• Remark 7
• Remark 8
• Remark 9