Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Long time asymptotics for the KPII equation

Derchyi Wu

Abstract

The long-time asymptotics of small Kadomtsev-Petviashvili II (KPII) solutions is derived using the inverse scattering theory and the stationary phase method.

Long time asymptotics for the KPII equation

Abstract

The long-time asymptotics of small Kadomtsev-Petviashvili II (KPII) solutions is derived using the inverse scattering theory and the stationary phase method.

Paper Structure

This paper contains 16 sections, 34 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. The IST for KPII equations
  3. The stationary points
  4. Long time asymptotics of $u_1(x)$
  5. Long time asymptotics of the eigenfunction for $u_{2,0}(x)$
  6. Representation formulas of the Cauchy integrals
  7. Asymptotics of the Cauchy integrals
  8. Long time asymptotics of $u_{2,0}(x)$ when $a>+\frac{1}{C}>0$
  9. Long time asymptotics of $u_{2,0}(x)$ when $a<-\frac{1}{C}<0$
  10. Long time asymptotics of the eigenfunction for $u_{2,1}(x)$
  11. Representation formulas of the Cauchy integrals
  12. Asymptotics of the Cauchy integrals
  13. Long time asymptotics of $u_{2,1}(x)$ when $a>+\frac{1}{C}>0$
  14. Long time asymptotics of $u_{2,1}(x)$ when $a<-\frac{1}{C}<0$
  15. A technical lemma
  16. ...and 1 more sections

Key Result

Theorem 1

Let $a=\pm 3r^2=\frac{x_2^2-3x_1x_3}{3x_3^2}$, $r>0$, and $t=-x_3$. Suppose Then, the solution $u$ to the Cauchy problem for E:KPII-intro with initial data $u_0$ satisfies : as $t\to +\infty$, Here, $s_c(\lambda)$ denotes the scattering data of $u_0$, $a$ characterizes the stationary points in the phase, and $t$ corresponds to the direction of KPII propagation.

Theorems & Definitions (63)

  • Theorem 1
  • Theorem 2: The Cauchy Problem W87
  • Definition 1
  • Lemma 3.1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 53 more