1. Introduction
2. Inverse scattering method for mKdV equation
2.1. Symmetry
2.1.1. Time evolution
2.2. Inverse scattering transformation
3. Robust inverse scattering and higher order soliton and rational traveling waves
Figure 1. Definition of the regions ${D}_{0},{D}_{\pm }$ and the contour ${{\rm{\Sigma }}}_{0}={{\rm{\Sigma }}}_{+}+{{\rm{\Sigma }}}_{-}.$ |
3.1. Darboux transformation in the robust inverse scattering transformation
If the spectral parameter ${\lambda }_{1}$ in the Darboux transformation (
From the definition of ${{\boldsymbol{\psi }}}_{s,r}^{\text{in}},$ we know
Setting ${\lambda }_{1}={\rm{i}}$ and ${\boldsymbol{c}}={\left[1,-1\right]}^{{\rm{T}}}$ in the zero background, we can get the second order soliton solution as
As to the nonzero constant background, we still set ${\lambda }_{1}={\rm{i}}$ and ${\boldsymbol{c}}={\left[1,-1\right]}^{{\rm{T}}},$ then the first order rational traveling waves is
3.2. The Riemann–Hilbert problem of higher order soliton and rational traveling waves
• | Analyticity: ${{\boldsymbol{M}}}_{s}(\lambda ;x,t)$ is analytic in $\lambda \in {\mathbb{C}}\backslash \partial {D}_{0}.$ |
• | Jump condition: ${{\boldsymbol{M}}}_{s}(\lambda ;x,t)$ takes the continuous boundary values from the exterior and the interior of ${D}_{0},$ when $\lambda \in \partial {D}_{0},$ they are related by the following jump condition $\begin{eqnarray}\begin{array}{l}\begin{array}{l}{{\boldsymbol{M}}}_{s,+}(\lambda ;x,t)={{\boldsymbol{M}}}_{s,-}(\lambda ;x,t){{\rm{e}}}^{-{\rm{i}}{\theta }_{s}(\lambda ;x,t){\sigma }_{3}}\\ \times \,{{\boldsymbol{G}}}_{s}^{[N-1]}(\lambda ;0,0)\cdots {{\boldsymbol{G}}}_{s}^{[0]}(\lambda ;0,0){{\rm{e}}}^{{\rm{i}}{\theta }_{s}(\lambda ;x,t){\sigma }_{3}},\lambda \in {{\rm{\Sigma }}}_{+},\end{array}\\ \begin{array}{l}{{\boldsymbol{M}}}_{s,+}(\lambda ;x,t)={{\boldsymbol{M}}}_{s,-}(\lambda ;x,t){{\rm{e}}}^{-{\rm{i}}{\theta }_{s}(\lambda ;x,t){\sigma }_{3}}\\ \times \,{{\boldsymbol{G}}}_{s}^{[0]}{(\lambda ;0,0)}^{-1}\cdots {{\boldsymbol{G}}}_{s}^{[N-1]}{(\lambda ;0,0)}^{-1}{{\rm{e}}}^{{\rm{i}}{\theta }_{s}(\lambda ;x,t){\sigma }_{3}},\\ \lambda \in {{\rm{\Sigma }}}_{-}.\end{array}\end{array}\end{eqnarray}$ |
• | Normalization: When $\lambda \to \infty ,$ we have ${{\boldsymbol{M}}}_{s}(\lambda ;x,t)\to {\mathbb{I}}.$ and Riemann–Hilbert Problem 2 For $(x,t)\in {{\mathbb{R}}}^{2}$, seek a $2\times 2$ matrix function ${{\boldsymbol{M}}}_{r}(\lambda ;x,t)$, which has the following properties: |
• | Analyticity: ${{\boldsymbol{M}}}_{r}(\lambda ;x,t)$ is analytic in $\lambda \in {\mathbb{C}}\backslash \left(\partial {D}_{0}{\cup }^{\,}{{\rm{\Sigma }}}_{c}\right).$ |
• | Jump condition: ${{\boldsymbol{M}}}_{r}(\lambda ;x,t)$ takes the continuous boundary values from the exterior and the interior of ${D}_{0}$ and the left side and the right side of ${{\rm{\Sigma }}}_{c},$ when $\lambda \in \partial {D}_{0}{\cup }^{\,}{{\rm{\Sigma }}}_{c},$ they are related by the following jump condition $\begin{eqnarray}\begin{array}{l}\begin{array}{l}{{\boldsymbol{M}}}_{r,+}(\lambda ;x,t)={{\boldsymbol{M}}}_{r,-}(\lambda ;x,t){{\rm{e}}}^{-{\rm{i}}{\theta }_{r}(\lambda ;x,t){\sigma }_{3}}\\ \times \,{{\boldsymbol{G}}}_{r}^{[N-1]}(\lambda ;0,0)\cdots {{\boldsymbol{G}}}_{r}^{[0]}(\lambda ;0,0){\boldsymbol{E}}(\lambda ){{\rm{e}}}^{{\rm{i}}{\theta }_{r}(\lambda ;x,t){\sigma }_{3}},\\ \lambda \in {{\rm{\Sigma }}}_{+},\end{array}\\ \begin{array}{l}{{\boldsymbol{M}}}_{r,+}(\lambda ;x,t)={{\boldsymbol{M}}}_{r,-}(\lambda ;x,t){{\rm{e}}}^{-{\rm{i}}{\theta }_{r}(\lambda ;x,t){\sigma }_{3}}{\boldsymbol{E}}{(\lambda )}^{-1}\\ \times \,{{\boldsymbol{G}}}_{r}^{[0]}{(\lambda ;0,0)}^{-1}\cdots {{\boldsymbol{G}}}_{r}^{[N-1]}{(\lambda ;0,0)}^{-1}{{\rm{e}}}^{{\rm{i}}{\theta }_{r}(\lambda ;x,t){\sigma }_{3}},\\ \lambda \in {{\rm{\Sigma }}}_{-}.\end{array}\end{array}\end{eqnarray}$ |
• | Normalization: When $\lambda \to \infty ,$ we have ${{\boldsymbol{M}}}_{r}(\lambda ;x,t)\to {\mathbb{I}}.$ |
According to the normalization principle about the Darboux transformation in the interior of ${D}_{0},$ we know that ${{\boldsymbol{G}}}_{s,r}^{[j]}(\lambda ;0,0)={{\boldsymbol{G}}}_{s,r}^{[0]}(\lambda ;0,0),j=1,2,\cdots ,N-1.$
3.3. Asymptotic analytic when $t$ is large
Figure 2. (a)–(c) The second order, fourth order and sixth order soliton for equation ( |
Figure 3. The upper ones are the odd-order rational traveling waves and the lower ones are the even-order rational waves for equation ( |
Following this rule of characteristic curves, for a general $2N$-order soliton, the characteristic curves can be set as
Figure 4. (a)–(c) The comparison between the exact soliton solution and the asymptotic solution, when the corresponding order is second, fourth and sixth respectively. The blue solid line is the exact soliton, and the red dotted line is the asymptotic soliton. |