Coherent structure of Alice-Bob modified Korteweg de-Vries Equation

Abstract

To describe two-place events, Alice-Bob systems have been established by means of the shifted parity and delayed time reversal in Ref. [1]. In this paper, we mainly study exact solutions of the integrable Alice-Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation are constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.

Figures (8)

• Figure 1: Plot of the single soliton \ref{['A1']} with $\mu_{1}=0.2$, $\nu_{1}=0.5$, $x_{0}=2$ and $t_{0}=2$ for the quantity $AB$.
• Figure 2: Display of the structure of the kink shape complexiton solution \ref{['34']} with $\mu_{1}=0.1$, $\nu_{1}=0.1$ and $x_{0}=t_{0}=0$ for (a) the real part $Re(A)=\Re(A)$, (b) the imaginary part $Im(A)=\Im(A)$ and (c) the amplitude $M(A)$.
• Figure 3: Rogue-wave solution \ref{['Rogue']} with $\rho=1$, $\kappa=i$, $x_{0}=0$ and $t_{0}=0$. Its density plot and shape are described in (a) and (b) respectively for the quantity $AB$.
• Figure 4: Rogue-wave solution \ref{['RW']} with the parameter selections \ref{['ccR']}. (a) Real part, (b) imaginary part, (c) amplitude and (d) the density plot of (c).
• Figure 5: The density plot of the interaction between a soliton (analytic) and a singular periodic wave described by \ref{['2s']} with the parameter selections $\lambda_1=2,\ \lambda_2=2.1,\ \lambda_3=2i,\ \lambda_4=-2i,\ x_0=t_0=0$.