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Application of the Riemann–Hilbert method to the vector modified Korteweg-de Vries equation

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Abstract

Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable systems. For the direct scattering problem, the spectral analysis is performed for the equation, from which a Riemann–Hilbert problem is well constructed. For the inverse scattering problem, the Riemann–Hilbert problem corresponding to the reflection-less case is solved. Furthermore, as applications, three types of multi-soliton solutions are found. Finally, some figures are presented to discuss the soliton behaviors of the vmKdV equation.

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xiu-Bin Wang & Bo Han

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  1. Xiu-Bin Wang
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  2. Bo Han
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Correspondence to Xiu-Bin Wang.

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This work is supported by the National Natural Science Foundation of China (NNSFC) under Grant No. 11871180.

Appendices

Appendix A

The \({\mathcal {N}}_{1}\)-soliton solution for the vmKdV equation (1) yields

$$\begin{aligned} \left\{ \begin{aligned}&q_{1}=2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha ^{(1)}_{k}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha ^{(1)}_{k}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha ^{(1)*}_{k-{\mathcal {N}}_{1}}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha ^{(1)*}_{k-{\mathcal {N}}_{1}}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj},\\&q_{2}=2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha ^{(2)}_{k}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha ^{(2)}_{k}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha ^{(2)*}_{k-{\mathcal {N}}_{1}}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha ^{(2)*}_{k-{\mathcal {N}}_{1}}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj},\\&\vdots \\&q_{n}=2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha ^{(n)}_{k}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha ^{(n)}_{k}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha ^{(n)*}_{k-{\mathcal {N}}_{1}}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha ^{(n)*}_{k-{\mathcal {N}}_{1}}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}, \end{aligned} \right. \end{aligned}$$
(62)

in which \(M=(m_{kj})_{2{\mathcal {N}}_{1}\times 2{\mathcal {N}}_{1}}\) with

$$\begin{aligned} m_{kj}=\left\{ \begin{aligned}&\frac{\left( \alpha ^{(1)*}_{k}\alpha ^{(1)}_{j}+\alpha ^{(2)*}_{k}\alpha ^{(2)}_{j}+\ldots +\alpha ^{(n)*}_{k}\alpha ^{(n)}_{j}\right) e^{\varTheta _{k}^{*}+\varTheta _{j}}+e^{-\varTheta _{k}^{*}-\varTheta _{j}}}{\lambda _{j}-\lambda _{k}^{*}},~~1\le k,j\le {\mathcal {N}}_{1};\\&\frac{\left( \alpha ^{(1)*}_{k}\alpha ^{(1)*}_{j-{\mathcal {N}}_{1}}+\alpha ^{(2)*}_{k}\alpha ^{(2)*}_{j-{\mathcal {N}}_{1}} +\ldots +\alpha ^{(n)*}_{k}\alpha ^{(n)*}_{j-{\mathcal {N}}_{1}}\right) e^{\varTheta _{k}^{*}+\varTheta _{j-{\mathcal {N}}_{1}}^{*}}+e^{-\varTheta _{k}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}^{*}}}{-\lambda _{j-{\mathcal {N}}_{1}}^{*}-\lambda _{k}^{*}},\\&~~1\le k\le {\mathcal {N}}_{1},~{\mathcal {N}}_{1}+1\le j\le 2{\mathcal {N}}_{1};\\&\frac{\left( \alpha ^{(1)}_{k-{\mathcal {N}}_{1}}\alpha ^{(1)}_{j}+\alpha ^{(2)}_{k-{\mathcal {N}}_{1}}\alpha ^{(2)}_{j} +\ldots +\alpha ^{(n)}_{k-{\mathcal {N}}_{1}}\alpha ^{(n)}_{j}\right) e^{\varTheta _{k-{\mathcal {N}}_{1}}+\varTheta _{j}}+e^{-\varTheta _{k-{\mathcal {N}}_{1}}-\varTheta _{j}}}{\lambda _{j}+\lambda _{k-{\mathcal {N}}_{1}}},\\&~~{\mathcal {N}}_{1}+1\le k\le 2{\mathcal {N}}_{1},~1\le j\le {\mathcal {N}}_{1};\\&\frac{\left( \alpha ^{(1)}_{k-{\mathcal {N}}_{1}}\alpha ^{(1)*}_{j-{\mathcal {N}}_{1}}+\alpha ^{(2)}_{k-{\mathcal {N}}_{1}}\alpha ^{(2)*}_{j-{\mathcal {N}}_{1}}+\ldots +\alpha ^{(n)}_{k-{\mathcal {N}}_{1}} \alpha ^{(n)*}_{j-{\mathcal {N}}_{1}}\right) e^{\varTheta _{k-{\mathcal {N}}_{1}}+\varTheta _{j-{\mathcal {N}}_{1}}^{*}}+e^{-\varTheta _{k-{\mathcal {N}}_{1}}-\varTheta _{j-{\mathcal {N}}_{1}}^{*}}}{\lambda _{j-{\mathcal {N}}_{1}}^{*}-\lambda _{k-{\mathcal {N}}_{1}}}.\\&~~{\mathcal {N}}_{1}+1\le k,j\le 2{\mathcal {N}}_{1}. \end{aligned} \right. \end{aligned}$$
(63)

Appendix B

Then an \({\mathcal {N}}_{2}\)-soliton solution for the vmKdV equation (1) reads

$$\begin{aligned} \left\{ \begin{aligned}&q_{1}=2i\sum _{k=1}^{{\mathcal {N}}_{2}}\sum _{j=1}^{{\mathcal {N}}_{2}}\alpha ^{(1)}_{k}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj},\\&q_{2}=2i\sum _{k=1}^{{\mathcal {N}}_{2}}\sum _{j=1}^{{\mathcal {N}}_{2}}\alpha ^{(2)}_{k}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj},\\&\vdots \\&q_{n}=2i\sum _{k=1}^{{\mathcal {N}}_{2}}\sum _{j=1}^{{\mathcal {N}}_{2}}\alpha ^{(n)}_{k}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj}, \end{aligned} \right. \end{aligned}$$
(64)

in which \(M=(m_{kj})_{{\mathcal {N}}_{2}\times {\mathcal {N}}_{2}}\) is defined by

$$\begin{aligned} m_{kj}=\frac{\left( \alpha ^{(1)}_{k}\alpha ^{(1)}_{j}{+}\alpha ^{(2)}_{k} \alpha ^{(2)}_{j}{+}\ldots {+}\alpha ^{(n)}_{k}\alpha ^{(n)}_{j}\right) e^{ \varTheta _{k}{+}\varTheta _{j}} {+}e^{{-}\varTheta _{k}{-}\varTheta _{j}}}{\lambda _{j}{-}\lambda _{k}^{*}}. \end{aligned}$$
(65)

Appendix C

The \(({\mathcal {N}}_{1}+{\mathcal {N}}_{2})\)-soliton solution for the vmKdV equation (1) arrives at

$$\begin{aligned} \left\{ \begin{aligned}&q_{1}=2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k}^{(1)}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k}^{(1)}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k-{\mathcal {N}}_{1}}^{(1)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k-{\mathcal {N}}_{1}}^{(1)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k}^{(1)}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k-{\mathcal {N}}_{1}}^{(1)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k}^{(1)}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k}^{(1)}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=2{\mathcal {N}}_{1}+11}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k}^{(1)}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj},\\&q_{2}=2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k}^{(2)}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k}^{(2)}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k-{\mathcal {N}}_{1}}^{(2)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k-N_{2}}^{(1)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+N_{2}}\alpha _{k}^{(2)}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}+1}^{2{\mathcal {N}}_{1}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k-{\mathcal {N}}_{1}}^{(2)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k}^{(2)}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k}^{(2)}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k}^{(2)}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj},\\&\vdots \\&q_{n}=2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k}^{(n)}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k}^{(n)}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k-{\mathcal {N}}_{1}}^{(n)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k-{\mathcal {N}}_{1}}^{(1)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=1}^{{\mathcal {N}}_{1}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+N_{2}}\alpha _{k}^{(n)}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj} +2i\sum _{k={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k-{\mathcal {N}}_{1}}^{(n)*}e^{\varTheta _{k-{\mathcal {N}}_{1}}^{*}-\varTheta _{j}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j=1}^{{\mathcal {N}}_{1}}\alpha _{k}^{(n)}e^{\varTheta _{k}-\varTheta _{j}^{*}}\left( M^{-1}\right) _{kj} +2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j={\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}}\alpha _{k}^{(n)}e^{\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}}\left( M^{-1}\right) _{kj}\\&+2i\sum _{k=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\sum _{j=2{\mathcal {N}}_{1}+1}^{2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}}\alpha _{k}^{(n)}e^{\varTheta _{k}-\varTheta _{j}}\left( M^{-1}\right) _{kj}, \end{aligned} \right. \end{aligned}$$
(66)

where \(M=(m_{kj})_{(2{\mathcal {N}}_{1}+{\mathcal {N}}_{2})\times (2{\mathcal {N}}_{1}+{\mathcal {N}}_{2})}\) is given by

$$\begin{aligned} m_{kj}=\left\{ \begin{aligned}&\frac{\left( \alpha ^{(1)*}_{k}\alpha ^{(1)}_{j}+\alpha ^{(2)*}_{k}\alpha ^{(2)}_{j}+\ldots +\alpha ^{(n)*}_{k}\alpha ^{(n)}_{j}\right) e^{\varTheta _{k}^{*}+\varTheta _{j}}+e^{-\varTheta _{k}^{*}-\varTheta _{j}}}{\lambda _{j}-\lambda _{k}^{*}},~~1\le k,j\le {\mathcal {N}}_{1};\\&\frac{\left( \alpha ^{(1)*}_{k}\alpha ^{(1)*}_{j-{\mathcal {N}}_{1}}+\alpha ^{(2)*}_{k}\alpha ^{(2)*}_{j-{\mathcal {N}}_{1}} +\ldots +\alpha ^{(n)*}_{k}\alpha ^{(n)*}_{j-{\mathcal {N}}_{1}}\right) e^{\varTheta _{k}^{*}+\varTheta _{j-{\mathcal {N}}_{1}}^{*}}+e^{-\varTheta _{k}^{*}-\varTheta _{j-{\mathcal {N}}_{1}}^{*}}}{-\lambda _{j-{\mathcal {N}}_{1}}^{*}-\lambda _{k}^{*}},\\&~~1\le k\le {\mathcal {N}}_{1},~{\mathcal {N}}_{1}+1\le j\le 2{\mathcal {N}}_{1};\\&\frac{\left( \alpha ^{(1)}_{k-{\mathcal {N}}_{1}}\alpha ^{(1)}_{j}+\alpha ^{(2)}_{k-{\mathcal {N}}_{1}}\alpha ^{(2)}_{j} +\ldots +\alpha ^{(n)}_{k-{\mathcal {N}}_{1}}\alpha ^{(n)}_{j}\right) e^{\varTheta _{k-{\mathcal {N}}_{1}}+\varTheta _{j}}+e^{-\varTheta _{k-{\mathcal {N}}_{1}}-\varTheta _{j}}}{\lambda _{j}+\lambda _{k-{\mathcal {N}}_{1}}},\\&~~{\mathcal {N}}_{1}+1\le k\le 2{\mathcal {N}}_{1},~1\le j\le {\mathcal {N}}_{1};\\&\frac{\left( \alpha ^{(1)}_{k-{\mathcal {N}}_{1}}\alpha ^{(1)*}_{j-{\mathcal {N}}_{1}}+\alpha ^{(2)}_{k-{\mathcal {N}}_{1}}\alpha ^{(2)*}_{j-{\mathcal {N}}_{1}}+\ldots +\alpha ^{(n)}_{k-{\mathcal {N}}_{1}} \alpha ^{(n)*}_{j-{\mathcal {N}}_{1}}\right) e^{\varTheta _{k-{\mathcal {N}}_{1}}+\varTheta _{j-{\mathcal {N}}_{1}}^{*}}+e^{-\varTheta _{k-{\mathcal {N}}_{1}}-\varTheta _{j-{\mathcal {N}}_{1}}^{*}}}{\lambda _{j-{\mathcal {N}}_{1}}^{*}-\lambda _{k-{\mathcal {N}}_{1}}},\\&~~{\mathcal {N}}_{1}+1\le k,j\le 2{\mathcal {N}}_{1};\\&\frac{\left( \alpha _{k-{\mathcal {N}}_{1}}^{(1)}\alpha _{j}^{(1)}+\alpha _{k-{\mathcal {N}}_{1}^{(2)}}\alpha _{j}^{(2)}+\ldots +\alpha _{k-{\mathcal {N}}_{1}}^{(n)}\alpha _{j}^{(n)}\right) e^{\varTheta _{k-{\mathcal {N}}_{1}}+\varTheta _{j}} +e^{-\varTheta _{k-{\mathcal {N}}_{1}}-\varTheta _{j}}}{\lambda _{j}+\lambda _{k-{\mathcal {N}}_{1}}},\\&~~{\mathcal {N}}_{1}+1\le k\le 2{\mathcal {N}}_{1},~2{\mathcal {N}}_{1}+1\le j\le 2{\mathcal {N}}_{1}+{\mathcal {N}}_{2};\\&\frac{\left( \alpha _{k}^{(1)}\alpha _{j}^{(1)}+\alpha _{k}^{(2)}\alpha _{j}^{(2)}+\ldots +\alpha _{k}^{(n)}\alpha _{j}^{(n)}\right) e^{\varTheta _{k}+\varTheta _{j}} +e^{-\varTheta _{k}-\varTheta _{j}}}{\lambda _{j}-\lambda _{k}^{*}},\\&~~2{\mathcal {N}}_{1}+1\le k\le 2{\mathcal {N}}_{1}+{\mathcal {N}}_{2},~1\le j\le {\mathcal {N}}_{1};\\&\frac{\left( \alpha _{k}^{(1)}\alpha _{j-{\mathcal {N}}_{1}}^{(1)*}+\alpha _{k}^{(2)}\alpha _{j-{\mathcal {N}}_{1}}^{(2)*}+\ldots +\alpha _{k}^{(n)}\alpha _{j-{\mathcal {N}}_{1}}^{(n)*}\right) e^{\varTheta _{k}+\varTheta _{j-{\mathcal {N}}_{1}}^{*}}+e^{-\varTheta _{k}-\varTheta _{j-{\mathcal {N}}_{1}}^{*}}}{-\lambda _{j-{\mathcal {N}}_{1}}^{*}-\lambda _{k}^{*}},\\&~~2{\mathcal {N}}_{1}+1\le k\le 2{\mathcal {N}}_{1}+N_{2},~~{\mathcal {N}}_{1}+1\le j\le 2{\mathcal {N}}_{1};\\&\frac{\left( \alpha _{k}^{(1)}\alpha _{j}^{(1)}+\alpha _{k}^{(2)}\alpha _{j}^{(2)}+\ldots +\alpha _{k}^{(n)}\alpha _{j}^{(n)}\right) e^{\varTheta _{j}+\varTheta _{k}} +e^{-\varTheta _{j}-\varTheta _{k}}}{\lambda _{j}-\lambda _{k}^{*}},\\&~~2{\mathcal {N}}_{1}+1\le k,j\le 2{\mathcal {N}}_{1}+{\mathcal {N}}_{2}. \end{aligned} \right. \end{aligned}$$
(67)

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Wang, XB., Han, B. Application of the Riemann–Hilbert method to the vector modified Korteweg-de Vries equation. Nonlinear Dyn 99, 1363–1377 (2020). https://doi.org/10.1007/s11071-019-05359-x

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