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Bifurcations of solitary waves in a coupled system of long and short waves

James Hornick, Dmitry E. Pelinovsky

Abstract

We consider families of solitary waves in the Korteweg--de Vries (KdV) equation coupled with the linear Schrödinger (LS) equation. This model has been used to describe interactions between long and short waves. To characterize families of solitary waves, we consider a sequence of local (pitchfork) bifurcations of the uncoupled KdV solitons. The first member of the sequence is the KdV soliton coupled with the ground state of the LS equation, which is proven to be the constrained minimizer of energy for fixed mass and momentum. The other members of the sequence are the KdV solitons coupled with the excited states of the LS equation. We connect the first two bifurcations with the exact solutions of the KdV--LS system frequently used in the literature.

Bifurcations of solitary waves in a coupled system of long and short waves

Abstract

We consider families of solitary waves in the Korteweg--de Vries (KdV) equation coupled with the linear Schrödinger (LS) equation. This model has been used to describe interactions between long and short waves. To characterize families of solitary waves, we consider a sequence of local (pitchfork) bifurcations of the uncoupled KdV solitons. The first member of the sequence is the KdV soliton coupled with the ground state of the LS equation, which is proven to be the constrained minimizer of energy for fixed mass and momentum. The other members of the sequence are the KdV solitons coupled with the excited states of the LS equation. We connect the first two bifurcations with the exact solutions of the KdV--LS system frequently used in the literature.

Paper Structure

This paper contains 18 sections, 12 theorems, 142 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence of traveling waves
  4. Variational characterization
  5. Schematic illustration and the road map
  6. Stability and bifurcations of the uncoupled KdV solitons
  7. Hessian operator for the solitary waves
  8. Characterization of the uncoupled KdV solitons
  9. Bifurcation of the coupled solitary wave at $\Omega_c^{(1)} \equiv \Omega_c$
  10. The bifurcating branch of coupled solitary waves
  11. Characterization of the bifurcating branch
  12. The Hessian operator for the exact solution (\ref{['solution-2']})
  13. Numerical approximation of $\langle g^2, L_1^{-1} g^2 \rangle$
  14. Bifurcation of the coupled solitary wave at $\Omega_c^{(2)}$
  15. The bifurcating branch of coupled solitary waves
  16. ...and 3 more sections

Key Result

Theorem 1

Assume $c > 0$, $\Omega < 0$, and $s = {\rm sgn}(k)$. If $s = -1$ or if $s = 1$ and $\Omega < \Omega_c$ with then the uncoupled soliton (solution-1) is a local minimizer of the constrained energy $H$ for fixed momentum $P$ degenerate only by the translational symmetry. If $s = 1$ and $\Omega \in (\Omega_c,0)$, then the uncoupled soliton (solution-1) is a saddle point of the constrained energy. Fu

Figures (4)

  • Figure 1: Schematic bifurcation diagram for $s = 1$ and $k = \frac{1}{2}$, which shows how the families of coupled solitary waves generalizing (\ref{['solution-2']}) and (\ref{['solution-3']}) are connected with the family (\ref{['solution-1']}) of the uncoupled KdV solitons. $\widehat{\mathcal{L}}$ denotes the Hessian operator constrained by two symmetries of the KdV--LS system (\ref{['KdV-NLS']}) and $n(\widehat{\mathcal{L}})$ is its Morse index.
  • Figure 2: Numerical approximation of $\int_{\mathbb{R}} g^2 W dy$ versus $p$.
  • Figure 3: The existence curve $\Omega_{\rm exact}(k,c)$ in (\ref{['Omega-third']}) (red) and the bifircation curve $\widetilde{\Omega}_c$ (black) versus $k \in \left(\frac{1}{6},\frac{1}{2}\right)$ for $c = 1$.
  • Figure 4: Numerical approximation of $\int_{\mathbb{R}} \tilde{g}^2 \tilde{W} dy$ versus $q$.

Theorems & Definitions (33)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • Lemma 2
  • Remark 6
  • ...and 23 more