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Long-time asymptotics of the Sawada-Kotera equation on the line

Deng-Shan Wang, Xiaodong Zhu

TL;DR

This work develops a comprehensive RH-based framework to analyze the long-time behavior of the Sawada–Kotera equation on the real line and its modified version via a 3×3 Lax pair. It establishes a Miura transformation connecting the SK and mSK RH problems, derives solitonless (radiation) long-time asymptotics, and partitions the (x,t) half-plane into distinct regions: decay, dispersive waves, a Painlevé-type self-similar regime, and rapid decay, with a transition region arising from the origin reflection data. In the Painlevé regime, the leading order is governed by a fourth-order analogue of Painlevé transcendents (the F-XVIII family) in a precisely constructed RH model, with explicit self-similar scalings. The paper also provides numerical verifications showing excellent agreement between the RH-based asymptotics and direct simulations, highlighting the utility of the RH-and-steepest-descent approach for higher-order integrable systems and extending KdV/KdV-type analyses to the SK/mSK hierarchy.

Abstract

The Sawada-Kotera (SK) equation is an integrable system characterized by a third-order Lax operator and is related to the modified Sawada-Kotera (mSK) equation through a Miura transformation. This work formulates the Riemann-Hilbert problem associated with the SK and mSK equations by using direct and inverse scattering transforms. The long-time asymptotic behaviors of the solutions to these equations are then analyzed via the Deift-Zhou steepest descent method for Riemann-Hilbert problems. It is shown that the asymptotic solutions of the SK and mSK equations are categorized into four distinct regions: the decay region, the dispersive wave region, the Painlevé region, and the rapid decay region. Notably, the Painlevé region is governed by the F-XVIII equation in the Painlevé classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlevé transcendents. This connection is established through the Riemann-Hilbert formulation in this work. Similar to the KdV equation, the SK equation exhibits a transition region between the dispersive wave and Painlevé regions, arising from the special values of the reflection coefficients at the origin. Finally, numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.

Long-time asymptotics of the Sawada-Kotera equation on the line

TL;DR

This work develops a comprehensive RH-based framework to analyze the long-time behavior of the Sawada–Kotera equation on the real line and its modified version via a 3×3 Lax pair. It establishes a Miura transformation connecting the SK and mSK RH problems, derives solitonless (radiation) long-time asymptotics, and partitions the (x,t) half-plane into distinct regions: decay, dispersive waves, a Painlevé-type self-similar regime, and rapid decay, with a transition region arising from the origin reflection data. In the Painlevé regime, the leading order is governed by a fourth-order analogue of Painlevé transcendents (the F-XVIII family) in a precisely constructed RH model, with explicit self-similar scalings. The paper also provides numerical verifications showing excellent agreement between the RH-based asymptotics and direct simulations, highlighting the utility of the RH-and-steepest-descent approach for higher-order integrable systems and extending KdV/KdV-type analyses to the SK/mSK hierarchy.

Abstract

The Sawada-Kotera (SK) equation is an integrable system characterized by a third-order Lax operator and is related to the modified Sawada-Kotera (mSK) equation through a Miura transformation. This work formulates the Riemann-Hilbert problem associated with the SK and mSK equations by using direct and inverse scattering transforms. The long-time asymptotic behaviors of the solutions to these equations are then analyzed via the Deift-Zhou steepest descent method for Riemann-Hilbert problems. It is shown that the asymptotic solutions of the SK and mSK equations are categorized into four distinct regions: the decay region, the dispersive wave region, the Painlevé region, and the rapid decay region. Notably, the Painlevé region is governed by the F-XVIII equation in the Painlevé classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlevé transcendents. This connection is established through the Riemann-Hilbert formulation in this work. Similar to the KdV equation, the SK equation exhibits a transition region between the dispersive wave and Painlevé regions, arising from the special values of the reflection coefficients at the origin. Finally, numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.

Paper Structure

This paper contains 23 sections, 41 theorems, 313 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. The modified Sawada-Kotera equation
  4. The Sawada-Kotera equation
  5. Numerical results
  6. Numerical verifications for the modified SK equation (\ref{['msk-equation']})
  7. Numerical results for SK equation
  8. The Riemman-Hilbert problem and Miura transformation
  9. Basic properties of the Jost functions
  10. The scattering matrix
  11. The eigenfunctions $M_n$
  12. Miura transformation between the SK equation and mSK equation
  13. Asymptotic analysis for Sectors $\rm I$ and $\rm II$
  14. Global parametrix $\bf \Delta$ and the first deformation
  15. The second deformation
  16. ...and 8 more sections

Key Result

Theorem 2.2

Suppose that $w_0(x) \in \mathcal{S}(\mathbb{R})$, then the reflection coefficients $\tilde{r}_1(k)$ and $\tilde{r}_2(k)$ are well-defined for $k \in{\Bbb R}_+$ and $k \in{\Bbb R}_-$, respectively, and satisfy the following properties:

Figures (18)

  • Figure 1: The asymptotic regions $\rm I$-$\rm IV$ in the $(x,t)$-half plane.
  • Figure 2: The asymptotic sectors $\rm I$-$\rm V$ in the $(x,t)$-half plane.
  • Figure 3: The comparisons of the leading-order asymptotic approximations in Theorem \ref{['msk Thm']} with the direct numerical simulations of the modified SK equation (\ref{['msk-equation']}) under initial Gaussian wave packet (\ref{['initial-msk-equation']}) at different time.
  • Figure 4: The comparisons of the leading-order asymptotic approximation from RH problem and direct numerical simulations of the SK equation (\ref{['SK']}) with initial data (\ref{['initial-SK-equation']}) at time $t= 50$ and $t=100$, respectively.
  • Figure 5: The contour $\Sigma$ decomposes the complex $k$-plane into six parts: $\Omega_n~(n=1,2,\cdots,6)$.
  • ...and 13 more figures

Theorems & Definitions (74)

  • Theorem 2.2
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.11
  • proof
  • Theorem 2.12
  • proof
  • ...and 64 more