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Prohibitions caused by nonlocality for Alice-Bob Boussinesq-KdV type systems

S. Y. Lou

Abstract

It is found that two different celebrate models, the Korteweg de-Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The model equation is called the Alice-Bob KdV (ABKdV) equation which was derived from the usual KdV equation via the so-called consistent correlated bang (CCB) companied by the shifted parity (SP) and delayed time reversal (DTR). The same model can be called as the Alice-Bob Boussinesq (ABB) system if the nonlocality is changed as only one of SP and DTR. For the ABB systems, with help of the bilinear approach and recasting the multi-soliton solutions of the usual Boussinesq equation to an equivalent novel form, the multi-soliton solutions with even numbers and the head on interactions are obtained. However, the multi-soliton solutions with odd numbers and the multi-soliton solutions with even numbers but with pursuant interactions are prohibited. For the ABKdV equation, the multi-soliton solutions exhibit many more structures because an arbitrary odd function of $x+t$ can be introduced as background waves of the usual KdV equation.

Prohibitions caused by nonlocality for Alice-Bob Boussinesq-KdV type systems

Abstract

It is found that two different celebrate models, the Korteweg de-Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The model equation is called the Alice-Bob KdV (ABKdV) equation which was derived from the usual KdV equation via the so-called consistent correlated bang (CCB) companied by the shifted parity (SP) and delayed time reversal (DTR). The same model can be called as the Alice-Bob Boussinesq (ABB) system if the nonlocality is changed as only one of SP and DTR. For the ABB systems, with help of the bilinear approach and recasting the multi-soliton solutions of the usual Boussinesq equation to an equivalent novel form, the multi-soliton solutions with even numbers and the head on interactions are obtained. However, the multi-soliton solutions with odd numbers and the multi-soliton solutions with even numbers but with pursuant interactions are prohibited. For the ABKdV equation, the multi-soliton solutions exhibit many more structures because an arbitrary odd function of can be introduced as background waves of the usual KdV equation.

Paper Structure

This paper contains 5 sections, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. AB integrable systems come from same equation with different non-localities
  3. Mult-soliton solutions for the ABB system (\ref{['ABKdV']}) with (\ref{['Bx']}) and/or (\ref{['Bt']})
  4. Multi-soliton solutions for the ABKdV system (\ref{['ABKdV']}) with (\ref{['Bxt']})
  5. summary and discussion

Figures (5)

  • Figure 1: Head on collision interaction between soliton and antisoliton for the field $A$ described by Eq. (\ref{['A']}) with \ref{['sol21']} under the parameter selections (\ref{['p2']}).
  • Figure 2: Head on collision of two paired soliton-antisolitons exhibited in Eq. (\ref{['A']}) with \ref{['f4']} and the parameter selections (\ref{['p4']}).
  • Figure 3: Plots of the head on collision interactions of six solitons expressed by Eq. (\ref{['A']}) with the parameter selections (\ref{['p6']}).
  • Figure 4: One soliton solution \ref{['NS']} with \ref{['FN']} for $N=k_1=1,x_0=t_0=0$ and the periodic background wave (\ref{['G1']}).
  • Figure 5: Plot of the interaction between the usual KdV soliton and the background few cycle soliton expressed by Eq. (\ref{['NS']}), \ref{['FN']} for $N=k_1=1,\ x_0=t_0=0$ and (\ref{['G2']}).