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Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation

Lorenzo Brandolese, Yuliia Samoilenko, Valerii Samoilenko

Abstract

The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization of the well known modified Camassa-Holm equation which is integrable system and in addition to the soliton solutions the equation has the peakon solutions. The novelty of the ideas of this paper lies in the development of a technique for constructing asymptotic peakon-like solutions. In the paper a general scheme of finding asymptotic approximation of any order is presented and accuracy of the asymptotic approximation is found. The obtained results are illustrated by examples both the soliton-like and the peakon-like solutions. For the examples the equations for the phase function as well as the main and the first terms of the soliton-like and peakon-like solutions are found. Moreover, for different values of a small parameter the graphs that demonstrate kind of the solutions are presented. The considered examples demonstrate that for an adequate description of the wave process it is enough obtain the main and the first terms of correspond asymptotic solutions. The results also confirm that the proposed technique can be used for constructing asymptotic wave-like solutions of other equations.

Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation

Abstract

The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization of the well known modified Camassa-Holm equation which is integrable system and in addition to the soliton solutions the equation has the peakon solutions. The novelty of the ideas of this paper lies in the development of a technique for constructing asymptotic peakon-like solutions. In the paper a general scheme of finding asymptotic approximation of any order is presented and accuracy of the asymptotic approximation is found. The obtained results are illustrated by examples both the soliton-like and the peakon-like solutions. For the examples the equations for the phase function as well as the main and the first terms of the soliton-like and peakon-like solutions are found. Moreover, for different values of a small parameter the graphs that demonstrate kind of the solutions are presented. The considered examples demonstrate that for an adequate description of the wave process it is enough obtain the main and the first terms of correspond asymptotic solutions. The results also confirm that the proposed technique can be used for constructing asymptotic wave-like solutions of other equations.
Paper Structure (15 sections, 7 theorems, 172 equations, 6 figures)

This paper contains 15 sections, 7 theorems, 172 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main definitions and form of the asymptotic solutions
  3. Algorithm of constructing asymptotic soliton-like solutions
  4. The main term of the asymptotic expansion
  5. The higher terms of the asymptotics
  6. Solvability of operator equation \ref{['equation_L']} in the space $\mathcal{S}(\mathbb{R})$
  7. Solvability of differential equation \ref{['sing_part_j2']} in the space ${\widetilde{G}}$
  8. Constructing terms of singular part of asymptotics
  9. Example
  10. The asymptotic peakon-like solutions
  11. Constructing asymptotic peakon-like solutions
  12. The main term
  13. Higher terms of the asymptotics
  14. Example
  15. Conclusions and discussions

Key Result

Theorem 1

Let the following conditions be fulfilled: Then, for any $t\in[0;T]$ equation equation_L has a solution $v(t,\cdot)$ in the space $\mathcal{S}(\mathbb{R})$ if and only if the function $\Phi$ satisfies the orthogonality condition of the form where the function $v_0$ is defined with formula v_0_1.

Figures (6)

  • Figure 1: The main term of the asymptotic soliton-like solution $V_0 (x, t, \tau)$, $\tau = (x - 6t) / \varepsilon$, as $\varepsilon=1$ (at the left) and $\varepsilon=0.5$ (at the right).
  • Figure 2: The term $\varepsilon V_1 (x, t, \tau)$, $\tau = (x - 6t) / \varepsilon$, as $\varepsilon=1$ (at the left) and $\varepsilon=0.5$ (at the right).
  • Figure 3: The first asymptotic approximation for soliton-like solution $u_1 (x, t, \varepsilon)$ as $\varepsilon=1$ (at the left) and $\varepsilon=0.5$ (at the right).
  • Figure 4: The main term of the asymptotic peakon-like solution $v_0 (t, \tau)$ as $\varepsilon=1$ (at the left) and $\varepsilon=0.5$ (at the right).
  • Figure 5: The term $\varepsilon V_1 (x, t, \tau)$ as $\varepsilon=1$ (at the left) and $\varepsilon=0.5$ (at the right).
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Theorem 2: GR1
  • proof : Proof of Theorem \ref{['theo1']}
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • Remark 3
  • ...and 7 more