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On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system

Kenneth Karlsen, Yan Rybalko

TL;DR

This work analyzes the Cauchy problem for a nonlocal, two-place FORQ equation by embedding it into a two-component peakon system and reducing the problem to an ODE in a Banach space. It proves local well-posedness in $H^s$ for $s>\frac{5}{2}$ and establishes continuous dependence of the solution on the initial data, including refined Hölder continuity results in a detailed Sobolev-index regime. The authors also reveal a bi-Hamiltonian structure linking the system to the AKNS hierarchy and discuss broader implications for nonlocal integrable systems. The methods provide a robust framework for small-time behavior of nonlocal peakon dynamics, with potential extensions to global behavior and singularity formation.

Abstract

We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic nonlinearity), we convert the Cauchy problem into a system of ordinary differential equations in a Banach space. Using this approach, we are able to demonstrate local well-posedness in the Sobolev space $H^{s}$ where $s > 5/2$. We also establish the continuity properties for the data-to-solution map for a range of Sobolev spaces. Finally, we briefly explore the relationship between the two-component system and the bi-Hamiltonian AKNS hierarchy.

On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system

TL;DR

This work analyzes the Cauchy problem for a nonlocal, two-place FORQ equation by embedding it into a two-component peakon system and reducing the problem to an ODE in a Banach space. It proves local well-posedness in for and establishes continuous dependence of the solution on the initial data, including refined Hölder continuity results in a detailed Sobolev-index regime. The authors also reveal a bi-Hamiltonian structure linking the system to the AKNS hierarchy and discuss broader implications for nonlocal integrable systems. The methods provide a robust framework for small-time behavior of nonlocal peakon dynamics, with potential extensions to global behavior and singularity formation.

Abstract

We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic nonlinearity), we convert the Cauchy problem into a system of ordinary differential equations in a Banach space. Using this approach, we are able to demonstrate local well-posedness in the Sobolev space where . We also establish the continuity properties for the data-to-solution map for a range of Sobolev spaces. Finally, we briefly explore the relationship between the two-component system and the bi-Hamiltonian AKNS hierarchy.
Paper Structure (11 sections, 20 theorems, 162 equations, 2 figures)

This paper contains 11 sections, 20 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Peakon system (\ref{['two-comp']}) and AKNS hierarchy
  4. Local existence and uniqueness
  5. Solution of the mollified system
  6. Existence of the solution of (\ref{['ODE']})
  7. Uniqueness of the solution of (\ref{['ODE']})
  8. Continuity of the data-to-solution map
  9. Hölder continuity properties of the solution
  10. Lipschitz continuity for $(s,r)\in A_1$ and $(s,p)\in B_1$
  11. Hölder continuity for $(s,r)\in A_i$ and $(s,p)\in B_j$, $i,j>1$

Key Result

Theorem 1.1

Consider the Cauchy problem for system two-comp with initial data $u_0(x)=u(x,0)$ and $v_0(x)=v(x,0)$, $x\in A$. Assume that $u_0,v_0\in H^s$, $s>\frac{5}{2}$. Then there exists a unique local solution $u,v\in C([-T_{\delta_0},T_{\delta_0}], H^{s}) \cap C^1([-T_{\delta_0},T_{\delta_0}], H^{s-1})$, as well as the size estimate for its derivative

Figures (2)

  • Figure 1: Regions $A_j$, $j=1,\dots,6$ in the $(s,r)$ plane.
  • Figure 2: Regions $B_j$, $j=1,\dots,6$ in the $(s,p)$ plane.

Theorems & Definitions (45)

  • Theorem 1.1: Local existence and uniqueness
  • Theorem 1.2: Continuity of the data-to-solution map
  • Remark 1.3: Nonuniform continuity of the data-to-solution map
  • Theorem 1.4: Hölder continuity of data-to-solution map
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.1
  • Lemma 2.2: Rellich-Kondrachov
  • proof
  • ...and 35 more