On the Kadomtsev-Petviashvili equation with double-power nonlinearities

TL;DR

This paper investigates the generalized Kadomtsev-Petviashvili equation with double-power nonlinearities, establishing existence, stability, instability, and blow-up criteria, and introduces novel minimization approaches without scaling assumptions.

Contribution

It introduces new minimization methods for ground state existence, extends stability analysis beyond convex Lyapunov functions, and provides numerical and theoretical insights into solution behaviors.

Findings

Existence of solitary waves under broad nonlinearities

Strong instability of ground states proven

Numerical exploration of blow-up and boundedness scenarios

Abstract

In this paper, we delve into the study of the generalized KP equation, which incorporates double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by $\mu_1|u|^{p_1-1}u+\mu_2|u|^{p_2-1}u$, with $p_1>p_2$, encompassing cases where $\mu_1>0$ and $\mu_1<0<\mu_2$. One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation.