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Long-time asymptotics of the KdV equation with delta function initial profile

Xuliang Liu, Deng-Shan Wang

TL;DR

This paper analyzes the long-time behavior of the KdV equation with singular, delta-function initial profiles using a Riemann–Hilbert problem and the Deift–Zhou nonlinear steepest descent method. It derives region-wise leading-order asymptotics, including a single soliton for delta wells, oscillatory dispersive tails, and Painlevé II self-similar behavior, with modifications for delta barriers that eliminate solitons. The authors extend the framework to multi-spike delta initial data, showing that 1 to L solitons can emerge depending on spike configuration and providing recursive constructions for the scattering data. The results demonstrate the effectiveness of the RH approach in capturing intricate long-time dynamics of integrable systems with highly singular initial data and offer precise, region-specific asymptotics that align with numerical observations.

Abstract

This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent method. When the initial value is a delta potential well, the asymptotic solution is predominantly dominated by a single soliton in certain region for $x>0$, while in other regions, the dispersive tails including self-similar region, collisionless shock region and dispersive wave region, play a more significant role. Conversely, when the initial value is a delta potential barrier, the soliton region is absent, although the dispersive tails still persist. Moreover, the general delta function initial profile with $L$-spikes is also studied and it is proved that one to $L$ solitons will be generated in soliton region, which depends on the sizes of the distance and height of the spikes. The leading-order terms of the solution in each region are derived, highlighting the efficacy of the Riemann-Hilbert formulation in elucidating the long-time behaviors of integrable systems.

Long-time asymptotics of the KdV equation with delta function initial profile

TL;DR

This paper analyzes the long-time behavior of the KdV equation with singular, delta-function initial profiles using a Riemann–Hilbert problem and the Deift–Zhou nonlinear steepest descent method. It derives region-wise leading-order asymptotics, including a single soliton for delta wells, oscillatory dispersive tails, and Painlevé II self-similar behavior, with modifications for delta barriers that eliminate solitons. The authors extend the framework to multi-spike delta initial data, showing that 1 to L solitons can emerge depending on spike configuration and providing recursive constructions for the scattering data. The results demonstrate the effectiveness of the RH approach in capturing intricate long-time dynamics of integrable systems with highly singular initial data and offer precise, region-specific asymptotics that align with numerical observations.

Abstract

This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent method. When the initial value is a delta potential well, the asymptotic solution is predominantly dominated by a single soliton in certain region for , while in other regions, the dispersive tails including self-similar region, collisionless shock region and dispersive wave region, play a more significant role. Conversely, when the initial value is a delta potential barrier, the soliton region is absent, although the dispersive tails still persist. Moreover, the general delta function initial profile with -spikes is also studied and it is proved that one to solitons will be generated in soliton region, which depends on the sizes of the distance and height of the spikes. The leading-order terms of the solution in each region are derived, highlighting the efficacy of the Riemann-Hilbert formulation in elucidating the long-time behaviors of integrable systems.

Paper Structure

This paper contains 18 sections, 22 theorems, 190 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Inverse Scattering Transform and RH Problem
  4. The Jost function for $u(x,0)=-U_0\delta(x)$
  5. Construct the Riemann-Hilbert problem
  6. Proofs of the Main Results
  7. Case I. The parameter $U_0>0$
  8. The soliton region and decay region
  9. Dispersive wave region
  10. Self-similar region
  11. Case II. The parameter $U_0<0$
  12. Dispersive wave region
  13. Self-similar region
  14. The general delta function initial profile with multi-spikes
  15. Potential of one delta function
  16. ...and 3 more sections

Key Result

Theorem 2.1

For $U_0>0$, let $\epsilon>0$ be small enough and $l>0$ be an integer, then as $t\to \infty$, the long-time asymptotic behaviors of the initial value problem $(InitialProblem)$ are formulated in the following forms (see Figure x0t-plane-U0>0)

Figures (17)

  • Figure 1: Regions for the long-time asymptotics of the KdV equation.
  • Figure 2: The comparisons of the leading-order terms in Theorem \ref{['theorem-1']} with the result of numerical simulation for $U_0=2$ and $t=50$, where the solid line represents the result of numerical simulation and the dashed lines represent the result of asymptotic analysis.
  • Figure 3: The individual comparisons of the leading-order terms in Theorem \ref{['theorem-1']} with the result of the numerical simulation in four different regions which are soliton region, self-similar region, collisionless shock region and dispersive wave region, where the parameter is $U_0=2$ and the time is $t=50$.
  • Figure 4: The comparisons of the leading-order terms in Theorem \ref{['theorem-1']} with the result of numerical simulation for $U_0=-2$ and $t=50$, where the solid line represents the result of numerical simulation and the dashed lines represent the result of asymptotic analysis.
  • Figure 5: The individual comparisons of the leading-order terms in Theorem \ref{['theorem-1']} with the result of the numerical simulation in four different regions which are dispersive wave region and self-similar region, where the parameter is $U_0=-2$ and the time is $t=50$.
  • ...and 12 more figures

Theorems & Definitions (34)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2: Scattering data
  • proof
  • Proposition 3.3: Time evolution of the scattering data
  • Proposition 3.4
  • ...and 24 more