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The "good" Boussinesq equation on the half-line: a Riemann-Hilbert approach

Christophe Charlier, Jonatan Lenells

Abstract

We consider the ``good" Boussinesq equation on the half-line. Assuming existence of the solution, we prove that it can be recovered from the solution of a $3\times 3$ Riemann-Hilbert problem that depends only on the initial and boundary values, and whose jump contour consists of twelve half-lines.

The "good" Boussinesq equation on the half-line: a Riemann-Hilbert approach

Abstract

We consider the ``good" Boussinesq equation on the half-line. Assuming existence of the solution, we prove that it can be recovered from the solution of a Riemann-Hilbert problem that depends only on the initial and boundary values, and whose jump contour consists of twelve half-lines.
Paper Structure (17 sections, 18 theorems, 80 equations, 2 figures)

This paper contains 17 sections, 18 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The direct problem
  4. Assumption of no solitons
  5. Assumption of generic behavior at $k = 0$
  6. Statement of the first theorem
  7. The inverse problem
  8. The RH problem for $M$
  9. Statement of the second theorem
  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. The spectral functions $s^A(k)$ and $S^A(k)$
  15. Proof of Theorem \ref{['r1r2th']}
  16. ...and 2 more sections

Key Result

Theorem 2.3

Suppose $u_0,v_0 \in \mathcal{S}({\Bbb R}_{+})$ and $\tilde{u}_{0}, \tilde{u}_{1}, \tilde{u}_{2}, \tilde{v}_0 \in C^{\infty}([0,T])$ are such that Assumption solitonlessassumption holds. Then the spectral functions $r_1:(0,\infty) \to {\Bbb C}$, $r_2:(-\infty,0) \to {\Bbb C}$ and $r_{3},r_{4}:\Gamma

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (23)

  • Theorem 2.3: Properties of $r_j(k)$, $j=1,2,3,4$
  • Definition 2.5
  • Theorem 2.6: Solution of (\ref{['boussinesqsystem']}) via inverse scattering
  • Corollary 2.8: Solution of (\ref{['boussinesqsystem']}) in terms of $n$
  • Remark 2.9
  • 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
  • Proposition 3.4: Asymptotics of $\mu_{j}$ as $k \to 0$
  • Proposition 3.5: Asymptotics of $\mu_{j}^{A}$ as $k \to 0$
  • ...and 13 more