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Lax Pairs: Integrable, Less Integrable and Nonintegrable Systems

D. C. Antonopoulou, S. Kamvissis

Abstract

Completely integrable finite dimensional Hamiltonian systems are well understood thanks to the work of Liouville and Arnold. On the other hand, the Lax Pair formulation of the KdV equation marks the beginning of the extension of the completely integrable theory to infinite dimensional Hamiltonian systems. Solutions of initial value problems for systems that admit a Lax Pair formulation normally have a tame qualitative behavior if Lax Pairs give rise to an infinite complete set of conserved laws. The situation is different for initial-boundary value problems, even in one space dimension. There are problems where integrability persists and regular (long time asymptotic) behavior can be proven (and we have proven them). There are others where even irregular "fractal-chaotic-looking" behavior can appear. In this short article we review an instance of each case. We also make a connection with results from the existing theory of perturbed Lax Pair equations on the real line.

Lax Pairs: Integrable, Less Integrable and Nonintegrable Systems

Abstract

Completely integrable finite dimensional Hamiltonian systems are well understood thanks to the work of Liouville and Arnold. On the other hand, the Lax Pair formulation of the KdV equation marks the beginning of the extension of the completely integrable theory to infinite dimensional Hamiltonian systems. Solutions of initial value problems for systems that admit a Lax Pair formulation normally have a tame qualitative behavior if Lax Pairs give rise to an infinite complete set of conserved laws. The situation is different for initial-boundary value problems, even in one space dimension. There are problems where integrability persists and regular (long time asymptotic) behavior can be proven (and we have proven them). There are others where even irregular "fractal-chaotic-looking" behavior can appear. In this short article we review an instance of each case. We also make a connection with results from the existing theory of perturbed Lax Pair equations on the real line.
Paper Structure (11 sections, 1 theorem, 26 equations, 5 figures)

This paper contains 11 sections, 1 theorem, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. NLS
  3. Long time asymptotics
  4. Sine-Gordon with Robin condition
  5. Comparison with the perturbed NLS on the real line
  6. Numerical approximation of the focusing NLS on the half line
  7. The numerical scheme
  8. Soliton starter
  9. Starters of larger measure
  10. Multiple solitons
  11. Conclusion. What next?

Key Result

Theorem 2.1

Let $q$ be the unique global classical solution $q\in C^1(L^2)\cap C^0(H^2)$ of the initial-value problem for defocusing NLS, with Dirichlet data $Q\in C^2$ and $Q(0)=q_0(0)$. Assume that $q_0\in H^1(0,\infty)\cap L^4(0,\infty)$ and $x q_0\in L^2(0,\infty)$. If $q(0,t)$, $q_t(0,t)$ have a sufficient Furthermore, if the Dirichlet data belong in the Schwartz class, then the Neumann data also belong

Figures (5)

  • Figure 6.1: Numerical solution's measure versus the exact solution's measure at $t=10$.
  • Figure 6.2: $|q(x,t)|$ at $t=10$.
  • Figure 6.3: $|q(x,t)|$ at $t=0,5,8$.
  • Figure 6.4: $|q(x,t)|$ at $t=0,5,8$.
  • Figure 6.5: $|q(x,t)|$ at $t=15$ when $c:=\frac{1}{5}$, $c:=\frac{1}{10}$, $c:=\frac{1}{15}$.

Theorems & Definitions (1)

  • Theorem 2.1