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On the $N_\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries

Weifang Weng, Zhenya Yan

Abstract

We explore the $N_{\infty}$-soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: $m_t+(m(u^2-u_x^2)^2)_x+κu_x=0,\quad m=u-u_{xx}$ with $\lim_{x\rightarrow \pm \infty }u(x,t)=0$. We mainly analyze the aggregation state of $N$-soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new $(y,t)$-space when the discrete spectra are located in different regions. Starting from the modified RH problem, we find that i) when the region is a quadrature domain with $\ell=n=1$, the corresponding $N_{\infty}$-soliton is the one-soliton solution which the discrete spectral point is the center of the region; ii) when the region is a quadrature domain with $\ell=n$, the corresponding $N_{\infty}$-soliton is an $n$-soliton solution; iii) when the discrete spectra lie in the line region, we provide its corresponding Riemann-Hilbert problem,; and iv) when the discrete spectra lie in an elliptic region, it is equivalent to the case of the line region.

On the $N_\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries

Abstract

We explore the -soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: with . We mainly analyze the aggregation state of -soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new -space when the discrete spectra are located in different regions. Starting from the modified RH problem, we find that i) when the region is a quadrature domain with , the corresponding -soliton is the one-soliton solution which the discrete spectral point is the center of the region; ii) when the region is a quadrature domain with , the corresponding -soliton is an -soliton solution; iii) when the discrete spectra lie in the line region, we provide its corresponding Riemann-Hilbert problem,; and iv) when the discrete spectra lie in an elliptic region, it is equivalent to the case of the line region.
Paper Structure (15 sections, 11 theorems, 102 equations, 4 figures)

This paper contains 15 sections, 11 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The modified Jost solutions
  4. The basic Riemann-Hilbert problem and $N$-soliton solution
  5. The modified Riemann-Hilbert problem
  6. $N_{\infty}$-soliton asymptotics: the quadrature domain
  7. $N_{\infty}$-soliton asymptotics: one-soliton solution
  8. $N_{\infty}$-soliton asymptotics: the n-soliton solution
  9. $N_{\infty}$-soliton asymptotics: the line domain
  10. Opening lenses
  11. The outer parametrix
  12. The local parametrix
  13. Small-norm Riemann-Hilbert problem
  14. $N_{\infty}$-soliton asymptotics: the elliptic domain
  15. Conclusions and discussions

Key Result

Lemma 4.1

For any open set $B_+$ containing the domain $\Omega_1$, the following identities hold: uniformly for all $\mathbb{C}\setminus B_+$. The boundary $\partial\Omega_1$ is counterclockwise.

Figures (4)

  • Figure 1: Distribution of discrete spectrum $K\cup K^*$ and the parameters are $s_1=\frac{3}{4}+i,s_2=\frac{1}{4},s_3=\frac{1}{3},m=1$.
  • Figure 2: Distribution of discrete spectrum for the line region.
  • Figure 3: The RH problem \ref{['RH9']} for the matrix function $N^{(4)}(y;z)$. Opening lenses $O_1$, $O_2$, $O_3$ and $O_4$.
  • Figure 4: Regional division for the Bessel model $U_{Bes}(z)$.

Theorems & Definitions (21)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 5.1
  • proof
  • ...and 11 more