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Superposition of shock waves of the generalized BBM equation

Alexey Samokhin

Abstract

The generalized BBM studied in this paper contains an additional dissipative term. Thus instead of solitons for the classic BBM there exists a lot of travelling shock wave solutions. The rules of their interactions or superposition is of high importance. The paper gives a detailed description of the two-parameter family of travelling wave solutions and proves their stability using a conservation law. Based on these results, effective rules of superposition are obtained. Moreover these rules are applicable not exclusively to the travelling wave solutions of BBM, but also to a wider class of shock waves, in particular discontinuous. Characteristic examples are illustrated by numerically worked out graphs.

Superposition of shock waves of the generalized BBM equation

Abstract

The generalized BBM studied in this paper contains an additional dissipative term. Thus instead of solitons for the classic BBM there exists a lot of travelling shock wave solutions. The rules of their interactions or superposition is of high importance. The paper gives a detailed description of the two-parameter family of travelling wave solutions and proves their stability using a conservation law. Based on these results, effective rules of superposition are obtained. Moreover these rules are applicable not exclusively to the travelling wave solutions of BBM, but also to a wider class of shock waves, in particular discontinuous. Characteristic examples are illustrated by numerically worked out graphs.
Paper Structure (6 sections, 3 theorems, 43 equations, 8 figures)

This paper contains 6 sections, 3 theorems, 43 equations, 8 figures.

Table of Contents

  1. Travelling wave shocks solutions
  2. The range of oscillations
  3. Stability of shock waves
  4. Superposition of shocks
  5. Examples
  6. Discussion

Key Result

Theorem 1

Every solution from $S_{H,h}$ tends to $T_{H,h}$ as $t\rightarrow\infty$.

Figures (8)

  • Figure 1: Left: Monotonic solution \ref{['4']}. Right: Its phase portrait. $(\lambda=0.5,\;\varepsilon=1,\; H=1.24,\;h=0.76)$
  • Figure 2: Left: Oscillatory solution: Right: Its phase portrait. $(\lambda=0.5,\;\varepsilon=1,\; H=4.24,\;h=-2)$
  • Figure 3: Types of TWS, medium: $\{\varepsilon=1, \lambda=0.5\}$.
  • Figure 4: Race results in turbulent union $T(4.56,-0.56)$, $V=3$.
  • Figure 5: Discontinuity results in a turbulent union $T(2.64,1.36)$, $V=3$.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Theorem 1: The stability property of travelling shock $T_{H,h}:$
  • proof
  • Corollary 1: Superposition rule.
  • proof
  • Remark 3
  • Corollary 2: Superposition velocity
  • proof
  • Remark 4
  • ...and 5 more