Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Boussinesq equation on the half-line

Christophe Charlier

TL;DR

This work extends the inverse scattering framework to the half-line for the Boussinesq equation via a $3 \times 3$ Lax pair, formulating a Riemann-Hilbert problem on a contour with 18 arcs, 18 segments, and 18 half-lines. The authors establish a direct problem that constructs nine spectral data functions from given initial and boundary values under solitonless and generic-k assumptions, and an inverse problem showing that the physical fields $u$ (and $v$) are recovered from the solution $M(x,t,k)$ of the RH problem. They further show an equivalent formulation in terms of a Zakharov-type system for $(u,v)$, and they provide detailed analyticity, symmetry, and asymptotic properties of the spectral data that underpin the RH construction. The approach yields a rigorous, reconstructive framework for the Boussinesq IBVP on the half-line, enabling theoretical analysis and potential long-time asymptotics within the Fokas method. Overall, the paper advances integrable PDE techniques for boundary value problems by delivering a concrete 3x3 RH formalism tied to the initial-boundary data via explicit spectral data relations.

Abstract

We study the initial-boundary value problem for the Boussinesq equation on the half-line. Assuming that the solution exists, we prove that it can be recovered from its initial-boundary values via the solution of a $3\times 3$ Riemann-Hilbert problem. The contour consists of $18$ arcs on the unit circle, $18$ segments and $18$ half-lines, and the associated jump matrices involve $9$ reflection coefficients.

The Boussinesq equation on the half-line

TL;DR

This work extends the inverse scattering framework to the half-line for the Boussinesq equation via a Lax pair, formulating a Riemann-Hilbert problem on a contour with 18 arcs, 18 segments, and 18 half-lines. The authors establish a direct problem that constructs nine spectral data functions from given initial and boundary values under solitonless and generic-k assumptions, and an inverse problem showing that the physical fields (and ) are recovered from the solution of the RH problem. They further show an equivalent formulation in terms of a Zakharov-type system for , and they provide detailed analyticity, symmetry, and asymptotic properties of the spectral data that underpin the RH construction. The approach yields a rigorous, reconstructive framework for the Boussinesq IBVP on the half-line, enabling theoretical analysis and potential long-time asymptotics within the Fokas method. Overall, the paper advances integrable PDE techniques for boundary value problems by delivering a concrete 3x3 RH formalism tied to the initial-boundary data via explicit spectral data relations.

Abstract

We study the initial-boundary value problem for the Boussinesq equation on the half-line. Assuming that the solution exists, we prove that it can be recovered from its initial-boundary values via the solution of a Riemann-Hilbert problem. The contour consists of arcs on the unit circle, segments and half-lines, and the associated jump matrices involve reflection coefficients.
Paper Structure (17 sections, 23 theorems, 114 equations, 5 figures)

This paper contains 17 sections, 23 theorems, 114 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The direct problem
  4. Assumption of no solitons
  5. Assumption of generic behavior at $k = \pm 1$
  6. Statement of the theorem
  7. The inverse problem
  8. An equivalent system
  9. Notation
  10. Spectral analysis
  11. Preliminaries
  12. The eigenfunctions $\mu_{1},\mu_{2},\mu_{3}$ and $\mu_{1}^{A},\mu_{2}^{A},\mu_{3}^{A}$
  13. The spectral functions $s(k)$ and $S(k)$
  14. Proof of Theorem \ref{['thm:r1r2']}
  15. The function $M$
  16. ...and 2 more sections

Key Result

Theorem 2.5

Suppose $u$ is a Schwartz class solution of (badboussinesq) with existence time $T \in (0, \infty)$, initial data $u_0, u_{1} \in \mathcal{S}({\Bbb R}_{+})$, and boundary values $\tilde{u}_{0},\tilde{u}_{1}, \tilde{u}_{2}, \tilde{u}_3 \in C^{\infty}([0,T])$ such that Assumptions solitonlessassumptio

Figures (5)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (36)

  • Definition 2.1
  • Remark 2.3
  • Theorem 2.5: Direct scattering: properties of $r_{1},\tilde{r}_{1},r_{2},\tilde{r}_{2},\hat{r}_{2},\check{r}_{2},R_{1}, R_{2}, \tilde{R}_{2}$
  • Theorem 2.7: Solution of (\ref{['badboussinesq']}) via inverse scattering
  • Definition 2.8
  • Lemma 2.9
  • proof
  • Proposition 3.1: Basic properties of $\mu_{1},\mu_{2},\mu_{3}$
  • Proposition 3.2: Basic properties of $\mu_{1}^{A},\mu_{2}^{A},\mu_{3}^{A}$
  • Proposition 3.3
  • ...and 26 more