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Notes on solitary-wave solutions of Rosenau-type equations

A. Durán, G. M. Muslu

Abstract

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration.

Notes on solitary-wave solutions of Rosenau-type equations

Abstract

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration.
Paper Structure (33 sections, 6 theorems, 145 equations, 15 figures, 4 tables)

This paper contains 33 sections, 6 theorems, 145 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Some mathematical properties of the Rosenau equations
  3. Existence of solitary wave solutions via Normal Form Theory
  4. Near $\mathcal{C}_{0}$
  5. Near $\mathcal{C}_{1}$
  6. Near $\mathcal{C}_{2}$
  7. Near $\mathcal{C}_{3}$
  8. Existence of CSW's with CC Theory
  9. Some examples
  10. Rosenau equation
  11. Rosenau-RLW equation
  12. Rosenau-KdV equation
  13. Rosenau-Kawahara equation
  14. Rosenau-RLW-Kawahara equation
  15. Concluding remarks
  16. ...and 18 more sections

Key Result

Theorem 2.1

Let $s\geq 0$. Assume that $\epsilon>0$, that (linearw) holds, and that $g$ in (GM1) is locally Lipschitz in $H^{s}$ with $g(0)=0$. Let $u_{0}\in H^{s}$. Then there exists $T>0$ and a unique solution $u\in X^{s}$ of (GM1) with initial condition $u_{0}$.

Figures (15)

  • Figure 1: Linearization at the origin of (\ref{['GM21a']}) (as in Figure 1 of Champ): Regions $1$ to $4$ in the $(b,a)$-plane, delimited by the bifurcation curves $\mathbb{C}_{0}$ to $\mathbb{C}_{3}$, and schematic representation of the position in the complex plane of the roots of (\ref{['GM22']}) for each curve and region. (Dot: simple root, larger dot: double root.)
  • Figure 2: CSW generation, $u$ profiles and phase portraits. (a), (b) Rosenau-RLW equation with $\alpha=-1, \epsilon=\beta=1, g(u)=u^{2}/2; c_{s}=1.1$; (c), (d) Rosenau-Kawahara equation with $\eta=-1/2, \epsilon=\beta=1, \gamma=2, g(u)=u^{2}/2, c_{s}=y_{-}+\epsilon+0.01\approx 0.951$; (e), (f) Rosenau-RLW-Kawahara equation with $\alpha=\gamma=-1, \epsilon=\beta=1, \eta=1, g(u)=u^{2}/2, c_{s}=z_{+}+\epsilon-0.565\approx 1.102$. The values of $y_{-}$ and $z_{+}$ are given in Tables \ref{['GMtav3']} and \ref{['GMtav4']} resp.
  • Figure 3: PTW generation, $u$ profiles and phase portraits. (a), (b) Rosenau-KdV equation with $\eta=1, \epsilon=\beta=1, g(u)=u^{2}/2; c_{s}=0.9$; (c), (d) Rosenau-Kawahara equation with $\eta=1, \epsilon=1, \beta=2, \gamma=1, g(u)=u^{2}/2, c_{s}=0.9$; (e), (f) Rosenau-RLW-Kawahara equation with $\alpha=\gamma=-1, \epsilon=\beta=1, \eta=1, g(u)=u^{2}/2, c_{s}=0.9$.
  • Figure 4: GSW generation, $u$ profiles and phase portraits. (a), (b) Rosenau-RLW equation with $\alpha=1, \epsilon=\beta=1, g(u)=u^{2}/2; c_{s}=0.9$; (c), (d) Rosenau-Kawahara equation with $\eta=1, \epsilon=0.25, \beta=4, \gamma=2, g(u)=u^{2}/2, c_{s}=0.43$; (e), (f) Rosenau-Kawahara equation with $\eta=-1, \epsilon=1, \beta=2, \gamma=1, g(u)=u^{2}/2, c_{s}=0.9$.
  • Figure 5: PTW generation, $u$ profiles and phase portraits. (a), (b) Rosenau-KdV equation with $\eta=-1, \epsilon=\beta=1, g(u)=u^{2}/2; c_{s}=1.05$; (c), (d) Rosenau-Kawahara equation with $\eta=1, \epsilon=0.25, \beta=4, \gamma=2, g(u)=u^{2}/2, c_{s}\approx 0.208$; (e), (f) Rosenau-Kawahara equation with $\alpha=0, \gamma=1, \epsilon=1/8, \beta=8, \eta=1, g(u)=u^{2}/2, c_{s}\approx 0.115$.
  • ...and 10 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 3 more