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

Luc Molinet, Stéphane Vento, Fred Weissler

Abstract

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

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

Abstract

We consider the Cauchy problem for the generalized fractional Korteweg-de Vries equation with homogeneous initial data . We show that, under smallness assumption on , and for a wide range of , including , we can construct a self-similar solution of this problem.

Paper Structure

This paper contains 7 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Notation and linear estimates
  3. Notation
  4. Linear estimates
  5. Construction of the profile
  6. Proof of Theorem \ref{['main-theorem']}
  7. Proof of Theorem \ref{['theo2']}

Key Result

Theorem 1

Let $1<\alpha\le 2$ and $p\ge 3$, $p \in \mathbb{N}$, with $p>\alpha/(\alpha-1)$, and let $\Phi$ be an homogeneous function of degree $-\alpha/p$ as in (Phi-form), with $|A_\alpha(1)\Phi|_{L^{p+1}}$ small enough, where $A_\alpha(1)$ is defined below in linear-group. Then there exists a profile $v\in

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Lemma 3
  • proof
  • ...and 3 more