Traveling waves and transverse instability for the fractional Kadomtsev-Petviashvili equation

TL;DR

This paper investigates traveling wave solutions of the fractional KP equation, proving existence of modulated solitary waves, analyzing their transverse stability, and performing numerical simulations for both fKP-I and fKP-II versions.

Contribution

It establishes the existence of modulated solitary waves via bifurcation and analyzes their transverse stability, including numerical validation for fractional KP equations.

Findings

Existence of periodically modulated solitary waves proved.

Line solitary waves of fKP-I are transversely linearly unstable.

Numerical experiments illustrate stability and instability dynamics.

Abstract

Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave solutions and their transverse (in)stability are discussed. Analogous to the classical Kadmomtsev--Petviashvili (KP) equation, the fKP equation comes in two versions: fKP-I and fKP-II. We show that the line solitary waves of fKP-I equation are transversely linearly instable. We also perform numerical experiments to observe the (in)stability dynamics of line solitary waves for both fKP-I and fKP-II equations.