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Journals >Acta Physica Sinica >Volume 69 >Issue 1 >Page 010204-1 > Article

Acta Physica Sinica
Vol. 69, Issue 1, 010204-1 (2020)

An integrable reverse space-time nonlocal Sasa-Satsuma equation

Cai-Qin Song¹ and Zuo-Nong Zhu^2、*

Author Affiliations

¹College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China

²School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

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DOI: 10.7498/aps.69.20191887 Cite this Article

Cai-Qin Song, Zuo-Nong Zhu. An integrable reverse space-time nonlocal Sasa-Satsuma equation[J]. Acta Physica Sinica, 2020, 69(1): 010204-1 Copy Citation Text

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Abstract

In this paper, we introduce an integrable reverse space-time nonlocal Sasa-Satsuma equation. The Darboux transformation and soliton solutions for this nonlocal integrable equation are constructed.

Keywords

Darboux transformation integrable reverse space-time nonlocal Sasa-Satsuma equation soliton solution

$ \begin{split} & {\rm i} Q_{T}+\frac{Q_{XX}}{2}+Q^2R\\ \, & + {\rm i}\left[Q_{XXX}+9QRQ_X+3Q^2R_X\right] = 0,\\ &{\rm i}R_{T}-\frac{R_{XX}}{2}-QR^2\\ \, & +{\rm i}\left[R_{XXX}+9QRR_X+3R^2Q_X\right] = 0, \end{split} $

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$ \begin{split} & {\rm i}Q_{T}+\frac{1}{2}Q_{XX}\pm |Q|^2Q\\ \, & +{\rm i} \left[Q_{XXX}\pm 9|Q|^2Q_X\pm 3|Q|^2Q^*_X)\right] = 0, \end{split} $

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$ {\rm i}q_t(x,t)+q_{xx}(x,t)\pm 2q^2(x,t)q^{*}(-x,t) = 0. $

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$ u_t+u_{xxx}\pm (9 uu_xu^*(-x,-t)+3u^2(u^*(-x,-t))_x) = 0. $

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$ \begin{split} & {\rm i}Q_{T}+\frac{Q_{XX}}{2}+Q^2Q(-X,-T)\\ \, & +{\rm i} \bigg[Q_{XXX} +9QQ(-X,-T)Q_X\bigg. \\\, & + \left. 3Q^2\frac{\partial Q(-X,-T)}{\partial X}\right] = 0.\end{split}$

(5)

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$\begin{split} & u(x,t) = Q(X,T)\exp\left\{\frac{-{\rm i}}{6}(X-\frac{T}{18})\right\}, \\ &t = T, x = X-\frac{T}{12} \end{split}$

(6)

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$ u_{t}+u_{xxx}+9uu(-x,-t)u_x+3u^2\frac{\partial u(-x,-t)}{\partial x} = 0. $

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$ \begin{split} & u_{t}+u_{xxx}+9uvu_x+3u^2v_x = 0,\\ & v_{t}+v_{xxx}+9uvv_x+3v^2u_x = 0, \end{split} $

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$ \begin{split} & u(x,t) = Q(X,T){\rm exp}\left\{\frac{-{\rm i}}{6}\left( {X - \frac{T}{{18}}} \right)\right\},\\ & v(x,t) = R(X,T){\rm exp}\left\{\frac{{\rm i}}{6}\left( {X - \frac{T}{{18}}} \right)\right\},\\ & t = T,\; \; x = X-\frac{T}{12} \end{split} $

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$ \begin{split}\, & { {\varPhi}}_x = { U}{ {\varPhi}},\quad { U} = {\rm i}\lambda { {\sigma}}_3 +{ P}, ~ { {\varPhi}}_t = { V}{ {\varPhi}}, \\ \, & { V} = 4{\rm i}\lambda^3 { {\sigma}}_3+4 \lambda^2 { P}+2{\rm i} \lambda({ P}^2+{ P}_x){ {\sigma}}_3\\ & \qquad +{ P}_x{ P}-{ {PP}}_x-{ {P}}_{xx}+2{ {P}}^3 \\[-10pt]\end{split} $

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$ { P} = \left( {\begin{array}{*{20}{c}} 0&0&u\\ 0&0&v\\ { - v}&{ - u}&0 \end{array}} \right),\;\;\;{{ {\sigma}} _3} = {\rm{diag}}(1,1, - 1). $

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$ { {\varTheta}}_j = (\phi_2(x,t,\lambda_j),\phi_1(x,t,\lambda_j),\phi_3(x,t,\lambda_j)) $

()

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$ { {\varTheta}}_x = -{ {\varTheta}} { U}, \; { {\varTheta}}_t = -{ {\varTheta}} { V} $

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$ M = \left( \begin{array}{*{20}{c}} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right). $

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$ \begin{array}{c} { {\varPhi}}[1] = { T}{ {\varPhi}} = { {\varPhi}}-{ {\eta}}_1{ {\varOmega}}({ {\eta}}_1,{ {\eta}}_1)^{-1}{ {\varOmega}}({ {\eta}}_1,{ {\varPhi}}), \end{array} $

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$\begin{split} & { {\varOmega}}({ {\eta}}_1,{ {\eta}}_1) = \left( {\begin{array}{*{20}{c}} \dfrac{{ {\varTheta}}_{1}{ {\varPhi}}_{1}}{-2\lambda_{1}} & \dfrac{{ {\varTheta}}_{1}{ {\varPhi}}_{2}}{-\lambda_{1}-\lambda_{2}} \\ \dfrac{{ {\varTheta}}_{2}{ {\varPhi}}_{1}}{-\lambda_{2}-\lambda_{1}} & \dfrac{{ {\varTheta}}_{2} { {\varPhi}}_{2}}{-2\lambda_{2}}\end{array}} \right),\\ & { {\varOmega}}({ {\eta}}_1,{ {\varPhi}}) = \left( {\begin{array}{*{20}{c}} \dfrac{{ {\varTheta}}_{1}{ {\varPhi}}}{-\lambda_{1}-\lambda} \\ \dfrac{{ {\varTheta}}_{2}{ {\varPhi}}}{-\lambda_{2}-\lambda} \end{array}} \right). \end{split}$

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$ \begin{array}{l} { {\varPhi}}[1]_x = { U}[1]{ {\varPhi}}[1],\quad { {\varPhi}}_x = { V}[1]{ {\varPhi}}[1], \end{array} $

(14)

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$ \begin{split} { U}[1] =\,& {\rm i} \lambda { {\sigma}}_3 +{ P}[1],\\ { V}[1] =\, & 4{\rm i}\lambda^3 { {\sigma}}_3+4 \lambda^2 { P}[1]+2{\rm i} \lambda({ P}[1]^2+{ P}[1]_x){ {\sigma}}_3\\ & +{ P}[1]_x{ P}[1]-{ P}[1]{ P}[1]_x-{ P}[1]_{xx}+2{ P}[1]^3. \end{split} $

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$ u[1] = u-2{\rm i}S_{13}, v[1] = v-2{\rm i}S_{23}, $

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$ { {\varPhi}}[n] = { {\varPhi}}-{ {RW}}^{-1}{ {\varOmega}}, $

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$\begin{split} & { W} = \left(\!\!\! \begin{array}{*{20}{c}} { {\varOmega}}({ {\eta}}_1,{ {\eta}}_1) & { {\varOmega}}({ {\eta}}_1,{ {\eta}}_2) &\cdots &{ {\varOmega}}({ {\eta}}_1,{ {\eta}}_n) \\ { {\varOmega}}({ {\eta}}_2,{ {\eta}}_1) & { {\varOmega}}({ {\eta}}_2,{ {\eta}}_2) & \cdots & { {\varOmega}}({ {\eta}}_2,{ {\eta}}_n) \\ \vdots& \vdots& \ddots & \vdots \\ { {\varOmega}}({ {\eta}}_n,{ {\eta}}_1) & { {\varOmega}}({ {\eta}}_n,{ {\eta}}_2) &\cdots& { {\varOmega}}({ {\eta}}_n,{ {\eta}}_n) \end{array}\!\!\!\right), \\ & { {\varOmega}} = \left(\!\!\begin{array}{*{20}{c}} { {\varOmega}}({ {\eta}}_1,{ {\varPhi}}) \\ { {\varOmega}}({ {\eta}}_2,{ {\varPhi}}) \\ \vdots \\ { {\varOmega}}({ {\eta}}_n,{ {\varPhi}}) \end{array} \!\right), \\[-35pt] \end{split}$

(18)

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$\begin{split} & { {\varOmega}}({ {\eta}}_k,{ {\eta}}_l) = \left(\!\!\!{\begin{array}{*{20}{c}} \dfrac{{ {\varTheta}}_{2k-1}{ {\varPhi}}_{2l-1}}{-\lambda_{2k-1}-\lambda_{2l-1}} & \dfrac{{ {\varTheta}}_{2k-1} { {\varPhi}}_{2l}}{-\lambda_{2k-1}-\lambda_{2l}} \\ \dfrac{{ {\varTheta}}_{2k}{ {\varPhi}}_{2l-1}}{-\lambda_{2k}-\lambda_{2l-1}} & \dfrac{{ {\varTheta}}_{2k}{ {\varPhi}}_{2l}}{-\lambda_{2k}-\lambda_{2l}} \end{array}}\!\!\! \right), \\ &\varOmega({ {\eta}}_k,{ {\varPhi}}) = \left(\!\!\!{\begin{array}{*{20}{c}}\dfrac{{ {\varTheta}}_{2k-1}{ {\varPhi}}}{-\lambda_{2k-1}-\lambda} \\ \dfrac{{ {\varTheta}}_{2k}{ {\varPhi}}}{-\lambda_{2k}-\lambda}\end{array}}\!\!\!\right).\end{split}$

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$ { P}[n] = { P}+{\rm i}[{ {RW}}^{-1}{ R}'{ M},{ {\sigma}}_3] $

(19)

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$ { a}W^{-1}{ b}' = \frac{\left| {\begin{array}{*{20}{c}} W & { b}' \\ -{ a} & 0 \end{array}} \right|}{|W|}, $

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$\begin{split} & u[n] = u-2{\rm i}\frac{\left| {\begin{array}{*{20}{c}} { W} & { r}'_3 \\ -{ r}_1 & 0 \end{array}} \right|}{|W|},\\ & v[n] = v-2{\rm i}\frac{\left| {\begin{array}{*{20}{c}} { W} & { r}'_3 \\ -{ r}_2 & 0 \end{array}} \right|}{|{ W}|}, \end{split}$

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$\begin{split} & \phi_{j1} = \alpha_j {\rm e}^{\theta_j},\;\; \phi_{j2} = \beta_j {\rm e}^{\theta_j},\;\; \phi_{j3} = {\rm e}^{-\theta_j}, \\ & \theta_j = {\rm i}\lambda_j(x+4\lambda_j^2t). \end{split} $

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$ u[1] = 4{\rm i}(\lambda_1+\lambda_2)\frac{g_1}{h},\; \; v[1] = 4{\rm i}(\lambda_1+\lambda_2)\frac{g_2}{h}, $

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$ \begin{split} h =\,& -2\lambda_1\lambda_2\left(1-2\alpha_1\beta_1{\rm e}^{4\theta_1}-2\alpha_2\beta_2 {\rm e}^{4\theta_2}\right.\\ & +4(\alpha_2\beta_1+\alpha_1\beta_2){\rm e}^{2(\theta_1+\theta_2)} \\ &+\left. 2{\rm e}^{4(\theta_1+\theta_2)}(\alpha_2^2\beta_1^2+\alpha_1^2\beta_2^2)\right)\\ &+(1+2\alpha_1\beta_1{\rm e}^{4\theta_1})(1+2\alpha_2\beta_2{\rm e}^{4\theta_2})(\lambda_1^2+\lambda_2^2),\\ g_1 = \,& (\alpha_1\lambda_1 {\rm e}^{2\theta_1}-\alpha_2\lambda_2 {\rm e}^{2\theta_2})(\lambda_1-\lambda_2)\\ &+2(\beta_1\alpha_2\lambda_1-\beta_2\alpha_1\lambda_2)(\alpha_2\lambda_1{\rm e}^{2\theta_2} \\ &-\alpha_1\lambda_2{\rm e}^{2\theta_1}){\rm e}^{2(\theta_1+\theta_2)},\\ g_2 = \,& (\beta_1\lambda_1 {\rm e}^{2\theta_1}-\beta_2\lambda_2 {\rm e}^{2\theta_2})(\lambda_1-\lambda_2)\\ &+2(\beta_1\alpha_2\lambda_1-\beta_2\alpha_1\lambda_2)(\beta_2\lambda_1{\rm e}^{2\theta_2} \\ &-\beta_1\lambda_2{\rm e}^{2\theta_1}){\rm e}^{2(\theta_1+\theta_2)}.\\[-10pt] \end{split} $

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Cai-Qin Song, Zuo-Nong Zhu. An integrable reverse space-time nonlocal Sasa-Satsuma equation[J]. Acta Physica Sinica, 2020, 69(1): 010204-1

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