Self-similar solutions for the generalized fractional Korteweg-de Vries equation

TL;DR

This paper constructs self-similar solutions for the generalized fractional Korteweg-de Vries equation, extending understanding of its behavior for various fractional orders and nonlinearities under small initial data.

Contribution

It introduces a method to construct self-similar solutions for a broad class of fractional KdV equations with different powers and fractional orders, including the case p=3.

Findings

Existence of self-similar solutions for a wide range of parameters.

Applicability to small initial data scenarios.

Extension to fractional orders up to 2.

Abstract

We consider the Cauchy problem for the generalized fractional Korteweg-de Vries equation $$ u_t+D^\alpha u_x + u^p u_x= 0, \quad 1<\alpha\le 2, \quad p\in {\mathbb N}\setminus\{0\}, $$ with homogeneous initial data $\Phi$. We show that, under smallness assumption on $\Phi$, and for a wide range of $(\alpha, p)$, including $p=3$, we can construct a self-similar solution of this problem.