Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-si
本站小编 Free考研考试/2022-01-02
Zu-feng Liang1, Xiao-yan Tang,2, Wei Ding31Department of Physics, Hangzhou Normal University, Hangzhou 10036, China 2School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, China 3School of Computer Science and Technology, Shanghai Normal University, Shanghai 200234, China
National Natural Science Foundation of China.11675055 National Natural Science Foundation of China.12071302
Abstract Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT. Keywords：Nonlocal conservation law;The modified KdV-sine-Gordon equation;Nonlocal symmetry
PDF (254KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Zu-feng Liang, Xiao-yan Tang, Wei Ding. Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation. Communications in Theoretical Physics, 2021, 73(5): 055003- doi:10.1088/1572-9494/abe9ad
1. Introduction
Symmetries and conservation laws of nonlinear partial differential equations (NPDEs) have been studied intensively and extensively, because they play an important role in the investigation of integrable properties, invariant solutions, stability analysis, numerical calculations and so on. The conservation law and symmetry can also be used to construct methods to explore nonlocally related PDE systems which is important in the analysis of a given PDE system [1]. The symmetry based method has been systematically shown to obtain non-invertible mappings of the Kolmogorov equation with variable coefficients to the backward heat equation, and the non-invertible mappings of linear hyperbolic PDEs with variable coefficients to linear hyperbolic PDEs with constant coefficients [2]. Recently, generalized symmetries, cosymmetries and local conservation laws of the isothermal no-slip drift flux model have been exhaustively described in [3]. The Lie symmetry analysis has also been used to study analytical solutions for time-fractional nonlinear systems such as the time-fractional Benjamin–Ono and Benjamin–Bona–Mahony equations with the Riemann–Liouville derivatives [4, 5].
Lately, nonlocal symmetries associated with linearizing transformations, Bäcklund transformations (BTs) and Darboux transformations have been studied a lot to construct highly nontrivial families of solutions, conservation laws and new integrable systems for some integrable systems [6–13]. For instance, the residual symmetry can be derived by the truncated Painlevé method and thus used to construct n-th Bäcklund transformations which lead to various solutions such as the lump and lump-type solutions [14, 15]. The nonlocal symmetries related to the BT have been studied for the sine-Gordon equation, and the associated topics on invariant solutions and nonlocal conservation laws have also been discussed [9, 10]. Here, we are concentrated on more nonlocal symmetries related to the BT for the modified KdV-sine-Gordon (mKdV-SG) equation$\begin{eqnarray}{u}_{{xt}}+\displaystyle \frac{3}{2}{u}_{x}^{2}{u}_{{xx}}+{u}_{{xxxx}}=\alpha \sin u,\end{eqnarray}$and some related topics including the nonlocal conservation laws, considering its important application in physics. The mKdV-SG equation (1) was first proposed when exploring nonlinear wave propagation in a monoatomic lattice where the anharmonic potential effect competes with the dispersive one under the influence of weak dislocation potential [16]. Then, in the study of optical pulse propagation in a medium described by a two-level Hamiltonian, it was rigorously demonstrated that the sG and mKdV equations can be derived from the Maxwell–Bloch equations by assuming the resonance frequency of the two-level atoms is either well above or well below the inverse of the characteristic duration of the pulse [17]. In the presence of both high- and low-frequency resonances, the nonlinear propagation of ultrashort pulses can be well described by the mKdV-SG equation. The mKdV-SG equation has also been mathematically investigated including its integrability property [18] and various soliton solutions [19–24].
Mathematically, the mKdV-SG equation (1) was shown to describe pseudospherical surfaces, namely, it is the integrability condition for the structural equation of such surfaces, and by means of a geometrical method, the BT of the mKdV-SG equation was obtained as [25],$\begin{eqnarray}{u}_{x}+{v}_{x}=2\eta \sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right),\end{eqnarray}$$\begin{eqnarray}\begin{array}{lcl}{u}_{t}-{v}_{t} & = & 2{f}_{32}-2{f}_{12}\cos \left(\displaystyle \frac{v}{2}-\displaystyle \frac{u}{2}\right)\\ & & +\,2{f}_{22}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right),\end{array}\end{eqnarray}$with$\begin{eqnarray}\begin{array}{rcl}{f}_{12} & = & \eta {v}_{{xx}}+\displaystyle \frac{\alpha }{\eta }\sin v,\\ {f}_{22} & = & \displaystyle \frac{\alpha }{\eta }\cos v-{\eta }^{3}-\displaystyle \frac{1}{2}\eta {v}_{x}^{2},\\ {f}_{32} & = & -{v}_{{xxx}}-{\eta }^{2}{v}_{x}-\displaystyle \frac{1}{2}{v}_{x}^{3}.\end{array}\end{eqnarray}$It means that if u is a solution of equation (1), then v determined by the BT (2)–(3) with (4) also satisfies the mKdV-SG equation in the form of$\begin{eqnarray}{v}_{{xt}}+\displaystyle \frac{3}{2}{v}_{x}^{2}{v}_{{xx}}+{v}_{{xxxx}}=\alpha \sin v.\end{eqnarray}$It is noted that a special nonlocal symmetry related to the BT (2)–(3) of equation (1) in its polynomial form has been obtained and some similarity solutions have been presented in [26].
The paper is arranged as follows. In the next section, three types of nonlocal symmetries of the mKdV-SG equation (1) associated with the BT (2)–(3) with (4) are obtained explicitly by requiring equations (1)–(4) form invariant under the infinitesimal transformations. Then infinitely many nonlocal conservation laws related to the nonlocal symmetries are obtained in section 3. Taking the nonlocal symmetries as symmetry constraint conditions applied on the BT, finite and infinite dimensional nonlinear systems are constructed in section 4. The last section is devoted to summary and discussions.
2. Nonlocal symmetries
Let us require the mKdV-SG equation (1) and its BT (2)–(3) are form invariant under the infinitesimal transformations $u\to u+\epsilon {\sigma }^{u},\quad v\to v+\epsilon {\sigma }^{v},\quad \eta \to \eta +\epsilon \lambda $, where ε is an infinitesimal parameter, σu, σv and λ are the symmetries of $u,\,v$ and η, respectively, then a linearized system can be derived as$\begin{eqnarray}\begin{array}{l}{\sigma }_{{xt}}^{u}+{\sigma }_{{xxxx}}^{u}+3{u}_{x}{u}_{{xx}}{\sigma }_{x}^{u}\\ +\displaystyle \frac{3}{2}{u}_{x}^{2}{\sigma }_{{xx}}^{u}-\alpha {\sigma }^{u}\cos u=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\sigma }_{{xt}}^{v}+{\sigma }_{{xxxx}}^{v}+3{v}_{x}{v}_{{xx}}{\sigma }_{x}^{v}\\ +\displaystyle \frac{3}{2}{v}_{x}^{2}{\sigma }_{{xx}}^{v}-\alpha {\sigma }^{v}\cos v=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\sigma }_{x}^{u}+{\sigma }_{x}^{v}-2\lambda \sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)\\ -\eta \cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)({\sigma }^{u}-{\sigma }^{v})=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\sigma }_{t}^{u}-{\sigma }_{t}^{v}-2{\sigma }_{{xxx}}^{v}-4\eta \lambda {v}_{x}+\displaystyle \frac{2\alpha \lambda }{{\eta }^{2}}\sin \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right)\\ \quad -\,\displaystyle \frac{\alpha }{\eta }({\sigma }^{u}+{\sigma }^{v})\cos \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right)\\ \ \ -(3{v}_{x}^{2}+2{\eta }^{2}){\sigma }_{x}^{v}-(2\eta {\sigma }_{{xx}}^{v}+2\lambda {v}_{{xx}}\\ \ \ -\,\left.\displaystyle \frac{\eta }{2}({v}_{x}^{2}+2{\eta }^{2})({\sigma }^{u}-{\sigma }^{v}\right)\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)\\ \ \ +\,(\eta {v}_{{xx}}({\sigma }^{u}-{\sigma }^{v})+2\eta {v}_{x}{\sigma }_{x}^{v}\\ \ \ +\,\lambda {v}_{x}^{2}+6\lambda {\eta }^{2})\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)=0.\end{array}\end{eqnarray}$It is shown that the symmetries σu, σv and λ satisfy a linear differential system of equations (6)–(9), however, it is still rather difficult to obtain its general solution. Therefore, we just write down three special solutions as below.
Case (1). The first type of nonlocal symmetry is obtained as$\begin{eqnarray}{\sigma }_{1}^{u}={{\rm{e}}}^{\eta p},\quad {\sigma }_{1}^{v}=0,\quad \lambda =0,\end{eqnarray}$with p given by$\begin{eqnarray}{p}_{x}=\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right),\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{p}_{t}=\displaystyle \frac{\alpha }{{\eta }^{2}}\cos \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right)-\displaystyle \frac{1}{2}({v}_{x}^{2}+2{\eta }^{2})\\ \quad \times \,\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)-{v}_{{xx}}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right).\end{array}\end{eqnarray}$It is easy to check that ${p}_{{xt}}={p}_{{tx}}$ is satisfied identically.
Case (2). The second type of nonlocal symmetry is found to be$\begin{eqnarray}{\sigma }_{2}^{u}=\lambda q{{\rm{e}}}^{\eta p},\quad {\sigma }_{2}^{v}=0,\end{eqnarray}$and λ is an arbitrary constant. Here, p satisfies equation (11), and q is determined by$\begin{eqnarray}{q}_{x}=2\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right){{\rm{e}}}^{-\eta p},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{q}_{t} & = & -\displaystyle \frac{2\alpha }{{\eta }^{2}}\sin \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right){{\rm{e}}}^{-\eta p}\\ & & -\,({v}_{x}^{2}+6{\eta }^{2})\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right){{\rm{e}}}^{-\eta p}\\ & & +\,2{v}_{{xx}}\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right){{\rm{e}}}^{-\eta p}+4\eta {v}_{x}{{\rm{e}}}^{-\eta p},\end{array}\end{eqnarray}$with ${q}_{{xt}}={q}_{{tx}}$ satisfied identically.
Case (3). The third type of nonlocal symmetry is given as$\begin{eqnarray}{\sigma }_{3}^{u}=f{{\rm{e}}}^{\eta p},\quad {\sigma }_{3}^{v}={v}_{x},\quad \lambda =0,\end{eqnarray}$where p satisfies equation (11), and f is determined by$\begin{eqnarray}{f}_{x}=-({v}_{{xx}}+\eta {p}_{x}{v}_{x}){{\rm{e}}}^{-\eta p},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{f}_{t} & = & ({v}_{{xxxx}}+2\eta {v}_{{xxx}}{p}_{x}+\left(\displaystyle \frac{3}{2}{v}_{x}^{2}+2{\eta }^{2}\right){v}_{{xx}}\\ & & +\,\eta {v}_{x}{p}_{t}+\eta {v}_{x}({v}_{x}^{2}+2{\eta }^{2}){p}_{x}+\alpha \sin v){{\rm{e}}}^{-\eta p},\end{array}\end{eqnarray}$and the consistent condition ${f}_{{xt}}={f}_{{tx}}$ is also satisfied identically.
It is remarkable that the above three special types of symmetries (10), (13) and (16) are called nonlocal because the function v in these symmetries is related to the function u through the BT (2)–(3). In addition, the first nonlocal symmetry (10) is equivalent to the nonlocal symmetry obtained in [26] for the mKdV-SG equation in its polynomial form. Likewise, we can make the special nonlocal symmetry (10) localized by introducing auxiliary functions and thus similarity solutions and similar interactive wave solutions can be obtained for the mKdV-SG equation (1).
Owing to the parameter η in these nonlocal symmetries σu, a series of infinitely many nonlocal symmetries can be generated straightforwardly. Here we would like to present a series of infinitely many nonlocal symmetries from the third nonlocal symmetry (16). Due to the procedure is routine as in [10, 11] and can be programmed directly, we just write down the final results. Substituting the following expansions$\begin{eqnarray}\begin{array}{rcl}p & = & \displaystyle \sum _{i=0}^{\infty }{p}_{i}{\delta }^{i},\quad f=\displaystyle \sum _{i=0}^{\infty }{f}_{i}{\delta }^{i},\\ v & = & \displaystyle \sum _{i=0}^{\infty }{v}_{i}{\delta }^{i},\quad {\sigma }_{3}^{u}=\displaystyle \sum _{i=0}^{\infty }{\sigma }_{3}^{u,i}{\delta }^{i},\end{array}\end{eqnarray}$where the expansion coefficients ${p}_{i},\,{v}_{i},\,{f}_{i}$ and ${\sigma }_{3}^{u,i}$ are functions of x and t, and δ is an arbitrary expansion constant, into the third nonlocal symmetry ${\sigma }_{3}^{u}$ in equation (16) together with equations (2), (3), (11), (12), (17), (18), and replacing η with η+δ, a series of infinitely many nonlocal symmetries can be obtained, where the first three are in the form of$\begin{eqnarray}{\sigma }_{3}^{u,1}={{\rm{e}}}^{\eta {p}_{0}}(\eta {p}_{1}{f}_{0}+{p}_{0}{f}_{0}+{f}_{1}),\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\sigma }_{3}^{u,2}=\displaystyle \frac{1}{2}{{\rm{e}}}^{\eta {p}_{0}}\left({f}_{0}{p}_{1}^{2}{\eta }^{2}+2({f}_{0}{p}_{0}{p}_{1}+{f}_{0}{p}_{2}+{f}_{1}{p}_{1})\eta \right.\\ \,\ \left.+\,{f}_{0}{p}_{0}^{2}+2{f}_{0}{p}_{1}+2{p}_{0}{f}_{1}+2{f}_{2}\right),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\sigma }_{3}^{u,3}=\displaystyle \frac{1}{6}{{\rm{e}}}^{\eta {p}_{0}}\left({f}_{0}{p}_{1}^{3}{\eta }^{3}+3{p}_{1}({f}_{0}{p}_{0}{p}_{1}\right.\\ \,\ +\,2{f}_{0}{p}_{2}+{f}_{1}{p}_{1}){\eta }^{2}+3({f}_{0}{p}_{0}^{2}{p}_{1}+2{f}_{0}{p}_{0}{p}_{2}\\ \qquad \ +2{f}_{0}{p}_{1}^{2}+2{f}_{1}{p}_{0}{p}_{1}+2{f}_{0}{p}_{3}+2{f}_{1}{p}_{2}+2{f}_{2}{p}_{1})\eta \\ \qquad \ +\,{f}_{0}{p}_{0}^{3}+3{f}_{1}{p}_{0}^{2}+6{f}_{0}{p}_{0}{p}_{1}\\ \left.\qquad \ +6{f}_{0}{p}_{2}+6{f}_{2}{p}_{0}+6{f}_{1}{p}_{1}+6{f}_{3}\right),\end{array}\end{eqnarray}$where the functions vi, ${p}_{i}\,(i=0,1,2,3)$ and f0 are determined by the following compatible conditions$\begin{eqnarray}{v}_{0x}+{u}_{x}-\eta \sin U=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{v}_{0x}^{3}+2{\eta }^{2}{v}_{0x}+2{v}_{0{xxx}}+{v}_{0t}-{u}_{t}\\ \ \ +\,2\eta {v}_{0{xx}}\cos U+\displaystyle \frac{2\alpha }{\eta }\sin W\\ \ \ -\,\eta ({v}_{0x}^{2}+2{\eta }^{2})\sin U=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{v}_{1x}+\eta {v}_{1}\cos U-2\sin U=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{v}_{1t}+(2{\eta }^{2}-\eta {v}_{0{xx}}{v}_{1}+{v}_{0x}^{2})\sin U\\ \ \ -\,\left(\displaystyle \frac{1}{2}{v}_{0x}^{2}{v}_{1}\eta +{\eta }^{3}{v}_{1}+2{v}_{0{xx}}\right)\cos U\\ \ \ -\,\displaystyle \frac{2\alpha }{{\eta }^{2}}\sin W+\displaystyle \frac{\alpha }{\eta }{v}_{1}\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{v}_{2x}+\displaystyle \frac{1}{4}\eta {v}_{1}^{2}\sin U+({v}_{1}+\eta {v}_{2})\cos U=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{v}_{2t}+2{v}_{0x}+\displaystyle \frac{{\eta }^{2}}{2}{v}_{0x}{v}_{1}^{2}-\left(\displaystyle \frac{3}{4}{\eta }^{3}{v}_{1}^{2}+\displaystyle \frac{1}{8}\eta {v}_{0x}^{2}{v}_{1}^{2}\right.\\ \ \ \left.+\,\eta {v}_{0{xx}}{v}_{2}-2\eta +{v}_{0{xx}}{v}_{1}\right)\sin U\\ \ \ -\,\left({\eta }^{3}{v}_{2}+3{\eta }^{2}{v}_{1}-\displaystyle \frac{1}{4}\eta {v}_{0{xx}}{v}_{1}^{2}\right.\\ \ \ \left.+\,\displaystyle \frac{1}{2}\eta {v}_{0x}^{2}{v}_{2}+\displaystyle \frac{1}{2}{v}_{0x}^{2}{v}_{1}\right)\cos U\\ \ \ -\,\displaystyle \frac{\alpha }{4{\eta }^{3}}({\eta }^{2}{v}_{1}^{2}-8)\sin W\\ \ \ -\,\displaystyle \frac{\alpha }{{\eta }^{2}}({v}_{1}-\eta {v}_{2})\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{v}_{3x}+\displaystyle \frac{1}{4}{v}_{1}(2\eta {v}_{2}+{v}_{1})\sin U\\ \ \ +\,(\eta {v}_{3}+{v}_{2}-\displaystyle \frac{1}{24}\eta {v}_{1}^{3})\cos U=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{v}_{3t}+\displaystyle \frac{1}{2}\eta {v}_{1}(2\eta {v}_{2}+{v}_{1}){v}_{0x}\\ \ \ -\,\left(\displaystyle \frac{3}{2}{\eta }^{3}{v}_{1}{v}_{2}+\displaystyle \frac{5}{4}{\eta }^{2}{v}_{1}^{2}\right.\\ \ \ +\,\displaystyle \frac{1}{4}\eta {v}_{0x}^{2}{v}_{1}{v}_{2}-\displaystyle \frac{1}{24}\eta {v}_{0{xx}}{v}_{1}^{3}+\eta {v}_{0{xx}}{v}_{3}\\ \ \ \left.+\displaystyle \frac{1}{8}{v}_{0x}^{2}{v}_{1}^{2}+{v}_{0{xx}}{v}_{2}-2\right)\sin U\\ \ \ +\,\left(\displaystyle \frac{1}{24}{\eta }^{3}{v}_{1}^{3}-{\eta }^{3}{v}_{3}-3{\eta }^{2}{v}_{2}+\displaystyle \frac{1}{48}\eta {v}_{1}^{3}{v}_{0x}^{2}-\displaystyle \frac{1}{2}\eta {v}_{0x}^{2}{v}_{3}\right.\\ \ \ \left.+\,\displaystyle \frac{1}{2}\eta {v}_{0{xx}}{v}_{1}{v}_{2}-3\eta {v}_{1}-\displaystyle \frac{1}{2}{v}_{0x}^{2}{v}_{2}+\displaystyle \frac{1}{4}{v}_{1}^{2}{v}_{0{xx}}\right)\cos U\\ \ \ -\,\displaystyle \frac{\alpha }{4{\eta }^{4}}(2{\eta }^{3}{v}_{1}{v}_{2}-{\eta }^{2}{v}_{1}^{2}+8)\sin W\\ \ \ -\,\displaystyle \frac{\alpha }{24{\eta }^{3}}({\eta }^{2}{v}_{1}^{3}-24{\eta }^{2}{v}_{3}\\ \ \ +\,24\eta {v}_{2}-24{v}_{1})\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{p}_{0x}-\cos U=0,\quad {p}_{0t}+{v}_{0{xx}}\sin U\\ \ \ +\,\displaystyle \frac{1}{2}({v}_{0x}^{2}+2{\eta }^{2})\cos U-\displaystyle \frac{\alpha }{{\eta }^{2}}\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{p}_{1x}-\displaystyle \frac{1}{2}{v}_{1}\sin U=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{p}_{1t}-\eta {v}_{0x}{v}_{1}+\displaystyle \frac{1}{4}{v}_{1}({v}_{0x}^{2}+6{\eta }^{2})\sin U\\ \ \ -\,\displaystyle \frac{1}{2}({v}_{0{xx}}{v}_{1}-4\eta )\cos U+\displaystyle \frac{\alpha }{2{\eta }^{2}}{v}_{1}\sin W\\ \ \ +\,\displaystyle \frac{2\alpha }{{\eta }^{3}}\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{p}_{2x}-\displaystyle \frac{1}{2}{v}_{2}\sin U+\displaystyle \frac{1}{8}{v}_{1}^{2}\cos U=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{p}_{2t}-\eta {v}_{2}{v}_{0x}+\displaystyle \frac{1}{8}(2{v}_{0x}^{2}{v}_{2}-{v}_{0{xx}}{v}_{1}^{2}\\ \ \ +\,12{\eta }^{2}{v}_{2}+8\eta {v}_{1})\sin U\\ \ \ +\,\displaystyle \frac{\alpha }{2{\eta }^{3}}(\eta {v}_{2}-2{v}_{1})\sin W\\ \ \ -\,\displaystyle \frac{1}{16}({v}_{0x}^{2}{v}_{1}^{2}+8{v}_{0{xx}}{v}_{2}+2{\eta }^{2}{v}_{1}^{2}-16)\cos U\\ \ \ +\,\displaystyle \frac{\alpha }{8{\eta }^{4}}({\eta }^{2}{v}_{1}^{2}-24)\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{p}_{3x}+\displaystyle \frac{1}{48}({v}_{1}^{3}-24{v}_{3})\sin U\\ \ \ +\,\displaystyle \frac{1}{4}{v}_{1}{v}_{2}\cos U=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{p}_{3t}+\displaystyle \frac{1}{24}{v}_{0x}\eta ({v}_{1}^{3}-24{v}_{3})\\ \ \\ -\,\left(\displaystyle \frac{1}{96}{v}_{0x}^{2}{v}_{1}^{3}-\displaystyle \frac{1}{4}{v}_{0x}^{2}{v}_{3}+\displaystyle \frac{1}{4}{v}_{0{xx}}{v}_{1}{v}_{2}\right.\\ \ \ +\,\displaystyle \frac{1}{16}{\eta }^{2}{v}_{1}^{3}-\displaystyle \frac{3}{2}{\eta }^{2}{v}_{3}-\eta {v}_{2}\\ \ \ \left.-\displaystyle \frac{1}{2}{v}_{1}\right)\sin U-\left(\displaystyle \frac{1}{8}{v}_{0x}^{2}{v}_{1}{v}_{2}-\displaystyle \frac{1}{48}{v}_{0{xx}}{v}_{1}^{3}\right.\\ \ \ \left.+\,\displaystyle \frac{1}{2}{v}_{0{xx}}{v}_{3}+\displaystyle \frac{1}{4}{\eta }^{2}{v}_{1}{v}_{2}+\displaystyle \frac{1}{4}\eta {v}_{1}^{2}\right)\cos U\\ \ \ -\,\displaystyle \frac{\alpha }{48{\eta }^{4}}({\eta }^{2}{v}_{1}^{3}-24{\eta }^{2}{v}_{3}+48\eta {v}_{2}-72{v}_{1})\sin W\\ \ \ +\,\displaystyle \frac{\alpha }{4{\eta }^{5}}({\eta }^{3}{v}_{1}{v}_{2}-{\eta }^{2}{v}_{1}^{2}+16)\cos W=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{f}_{0x}+{{\rm{e}}}^{-\eta {p}_{0}}(\eta {v}_{0x}\cos U+{v}_{0{xx}})=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{f}_{0t}-{{\rm{e}}}^{-\eta {p}_{0}}\left(\alpha \sin {v}_{0}+\displaystyle \frac{3}{2}{v}_{0x}^{2}{v}_{0{xx}}\right.\\ \ \ +\,2{\eta }^{2}{v}_{0{xx}}+{v}_{0{xxxx}}-\eta {v}_{0x}{v}_{0{xx}}\sin U\\ \ \ +\displaystyle \frac{1}{2}\eta ({v}_{0x}^{3}+2{\eta }^{2}{v}_{0x}+4{v}_{0{xxx}})\cos U\\ \ \ \left.+\,\displaystyle \frac{\alpha }{\eta }{v}_{0x}\cos W\right)=0,\end{array}\end{eqnarray}$with $U=\displaystyle \frac{1}{2}(u-{v}_{0}),\,W=\tfrac{1}{2}(u+{v}_{0})$. Here the conditions for ${f}_{i}\,(i=1,2,3)$ are not given explicitly as their expressions are too complicated. It is easy to check that all the compatibility conditions ${v}_{{ixt}}={v}_{{itx}}$, ${p}_{{ixt}}={p}_{{itx}}$ and ${f}_{{ixt}}={f}_{{itx}}$ are identically satisfied.
3. Infinitely many nonlocal conservation laws related to the nonlocal symmetries
To find conservation laws for the mKdV-SG equation is nothing but to find the pairs of ρ and J satisfying$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\rho (x,t)+\displaystyle \frac{\partial }{\partial x}J(x,t)=0,\end{eqnarray}$where ρ and J are called the conserved density and the conserved flux, respectively, for any solution u of equation (1). Many effective ways have been established to study conservation laws. Our previous work [10, 11] show that the divergence expression (40) can be obtained for infinitely many conservation laws by applying the parameter expansion method either to some auxiliary functions involved in solving the symmetry equations (8) and (9), or directly to the BT. Here, we are only concentrated on infinitely many nonlocal conservation laws corresponding to the nonlocal symmetries related to the BT.
Integrating equations (8) and (9) with respect to x and t, respectively, gives$\begin{eqnarray}\begin{array}{rcl}{\sigma }^{u} & = & -2\eta {{\rm{e}}}^{\eta p}\displaystyle \int {\sigma }^{v}{{\rm{e}}}^{-\eta p}\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right){\rm{d}}x\\ & & +\,(\lambda q+C){{\rm{e}}}^{\eta p}-{\sigma }^{v},\end{array}\end{eqnarray}$and$\begin{eqnarray}\begin{array}{rcl}{\sigma }^{u} & = & \displaystyle \frac{{{\rm{e}}}^{\eta p}}{\eta }\displaystyle \int {{\rm{e}}}^{-\eta p}\left[-2{\eta }^{2}{v}_{x}{\sigma }_{x}^{v}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)\right.\\ & & +\,2{\eta }^{2}{\sigma }_{{xx}}^{v}\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)+3\eta {v}_{x}^{2}{\sigma }_{x}^{v}+2{\eta }^{3}{\sigma }_{x}^{v}\\ & & \left.+\,2\alpha {\sigma }^{v}\cos \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right)+2\eta {\sigma }_{{xxx}}^{v}\right]\\ & & \times \,{\rm{d}}t+(\lambda q+C){{\rm{e}}}^{\eta p}+{\sigma }^{v},\end{array}\end{eqnarray}$where the arbitrary integration functions have been simplified to a same constant C, p and q are determined by equations (11)–(12) and equations (14)–(15), respectively. To equal equations (41) and (42) arrives at$\begin{eqnarray}\begin{array}{l}\displaystyle \int {{\rm{e}}}^{-\eta p}\left[2{\eta }^{2}{v}_{x}{\sigma }_{x}^{v}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)\right.\\ \ \ -\,2{\eta }^{2}{\sigma }_{{xx}}^{v}\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)-3\eta {v}_{x}^{2}{\sigma }_{x}^{v}\\ \ \ -\,2{\eta }^{3}{\sigma }_{x}^{v}-2\eta {\sigma }_{{xxx}}^{v}\\ \ \ \left.-2\alpha {\sigma }^{v}\cos \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right)\right]{\rm{d}}t-2{\eta }^{2}\displaystyle \int {\sigma }^{v}{{\rm{e}}}^{-\eta p}\\ \ \ \times \,\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right){\rm{d}}x-2\eta {\sigma }^{v}{{\rm{e}}}^{-\eta p}=0,\end{array}\end{eqnarray}$which leads to a nontrivial new nonlocal conservation law with the conserved density and flux as$\begin{eqnarray}\rho =2\eta {\sigma }_{x}^{v}{{\rm{e}}}^{-\eta p},\end{eqnarray}$and$\begin{eqnarray}\begin{array}{l}J=\left[2{\eta }^{2}{v}_{x}{\sigma }_{x}^{v}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)-2{\eta }^{2}{\sigma }_{{xx}}^{v}\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{v}{2}\right)\right.\\ \ \ -\,3\eta {v}_{x}^{2}{\sigma }_{x}^{v}-2{\eta }^{3}{\sigma }_{x}^{v}-2\eta {\sigma }_{{xxx}}^{v}\\ \ \ \left.-2\alpha {\sigma }^{v}\cos \left(\displaystyle \frac{u}{2}+\displaystyle \frac{v}{2}\right)\right]{{\rm{e}}}^{-\eta p},\end{array}\end{eqnarray}$respectively.
It is seen that the exponential part in this new nonlocal conservation law is related to the first nonlocal symmetry ${\sigma }_{1}^{u}$ given by equation (10), while σv is any symmetry of the function v in equation (5) determined by the system of equations (6)–(9). Consequently, the nonlocal conservation law is related to the nonlocal symmetries.
Moreover, a series of infinitely many nonlocal conservation laws, in the form of ${\partial }_{t}{\rho }_{1i}(x,t)+{\partial }_{x}{J}_{1i}(x,t)\,=0,\,i\,=\,0,1,2,\ldots $, can be obtained by applying the parameter expansion method, namely inserting the expansions of the functions $p,\,v$ in equation (19) into the conserved density (44), the flux (45) and the related equations (7)–(8), (11)–(12) with η replaced by η+δ, and then equalling zero the coefficients of the same orders of δ. For illustration and for simplicity, here we just write down the new nonlocal conserved density and flux for i 1 as$\begin{eqnarray}{\rho }_{1}=2\left[(1-\eta {p}_{0}-{\eta }^{2}{p}_{1}){\sigma }_{0x}^{v}+\eta {\sigma }_{1x}^{v})\right]{{\rm{e}}}^{\eta {p}_{0}},\end{eqnarray}$and the corresponding nonlocal conserved flux reads$\begin{eqnarray}\begin{array}{rcl}{J}_{1} & = & \left[\left(2\eta ({\eta }^{2}{p}_{1}+\eta {p}_{0}-2){\sigma }_{0{xx}}^{v}\right.\right.\\ & & \left.-\,{v}_{1}{v}_{0x}{\eta }^{2}{\sigma }_{0x}^{v}-2{\eta }^{2}{\sigma }_{1{xx}}^{v}\right)\cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{v}_{0}}{2}\right)\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}+2\alpha \left(\eta {p}_{1}{\sigma }_{0}^{v}+{p}_{0}{\sigma }_{0}^{v}-{\sigma }_{1}^{v}\right)\cos \left(\displaystyle \frac{u}{2}+\displaystyle \frac{{v}_{0}}{2}\right)\\ \ \ +\,\alpha {v}_{1}{\sigma }_{0}^{v}\sin \left(\displaystyle \frac{u}{2}+\displaystyle \frac{{v}_{0}}{2}\right)\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}+\left(2\eta ({v}_{1x}\eta -({\eta }^{2}{p}_{1}+\eta {p}_{0}-2){v}_{0x}){\sigma }_{0x}^{v}\right.\\ \ \ +\,\left.2{v}_{0x}{\eta }^{2}{\sigma }_{1x}^{v}-{v}_{1}{\eta }^{2}{\sigma }_{0{xx}}^{v}\right)\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{v}_{0}}{2}\right)\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}+\left(3({\eta }^{2}{p}_{1}+\eta {p}_{0}-1){v}_{0x}^{2}-6{v}_{1x}{v}_{0x}\eta \right.\\ \ \ +\,\left.2{\eta }^{2}({\eta }^{2}{p}_{1}+\eta {p}_{0}-3\right){\sigma }_{0x}^{v}\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}+2({\eta }^{2}{p}_{1}+\eta {p}_{0}-1){\sigma }_{0{xxx}}^{v}-2\eta {\sigma }_{1{xxx}}^{v}\\ \ \ -\,\left.\eta (3{v}_{0}^{2}+2{\eta }^{2}){\sigma }_{1x}^{v}\right]{{\rm{e}}}^{-\eta {p}_{0}},\end{array}\end{eqnarray}$where v0, v1, p0 and p1 are determined by equations (23)–(26), (31)–(33), ${\sigma }_{0}^{v}$ and ${\sigma }_{1}^{v}$ are determined by$\begin{eqnarray}\begin{array}{l}{\sigma }_{0{xt}}^{v}+{\sigma }_{0{xxxx}}^{v}+\displaystyle \frac{3}{2}{v}_{0x}^{2}{\sigma }_{0{xx}}^{v}\\ \ \ +\,3{\sigma }_{0x}^{v}{v}_{0x}{v}_{0{xx}}-\alpha {\sigma }_{0}^{v}\cos {v}_{0}=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\sigma }_{1{xt}}^{v}+{\sigma }_{1{xxxx}}^{v}+\displaystyle \frac{3}{2}{v}_{0x}^{2}{\sigma }_{1{xx}}^{v}+3({\sigma }_{1x}^{v}{v}_{0{xx}}\\ \ \ +\,{\sigma }_{0x}^{v}{v}_{1{xx}}+{v}_{1x}{\sigma }_{0{xx}}^{v}){v}_{0x}+3{\sigma }_{0x}^{v}{v}_{0{xx}}{v}_{1x}\\ \ \ +\,\alpha {\sigma }_{0}^{v}{v}_{1}\sin {v}_{0}-\alpha {\sigma }_{1}^{v}\cos {v}_{0}=0,\end{array}\end{eqnarray}$which are obtained in the same way by substituting the expansion of v in equation (19) and ${\sigma }^{v}={\sum }_{i=0}^{\infty }{\sigma }_{i}^{v}{\delta }^{i}$ into equation (7).
4. New integrable systems from nonlocal symmetries
In this section, we present some new nonlinear systems integrable in the sense of possessing infinitely many symmetries by means of the symmetry constraint method, namely, applying some nonlocal symmetry constraint conditions on the BT (2)–(3) of the mKdV-SG equation (1).
Case 1: One dimensional nonlinear integrable systems. First we impose the generalized symmetry constraint with the first nonlocal symmetry (10) on the x-part of the BT (54) as$\begin{eqnarray}{u}_{x}=\displaystyle \sum _{i=1}^{N}{a}_{i}\exp \left({\eta }_{i}\int \cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right){\rm{d}}x\right),\end{eqnarray}$then the first kind of the finite dimensional $(N+1)$-component integro-differential equations can be obtained as$\begin{eqnarray}{u}_{x}=\displaystyle \sum _{i=1}^{N}{a}_{i}\exp \left({\eta }_{i}\int \cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right){\rm{d}}x\right),\end{eqnarray}$$\begin{eqnarray}{u}_{x}+{u}_{{ix}}=2{\eta }_{i}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right),\quad i=1,2,...,N,\end{eqnarray}$with arbitrary constants ai and ηi. Introduce$\begin{eqnarray}{u}_{i}=u-2\arccos \displaystyle \frac{{\left(\mathrm{ln}{g}_{{ix}}\right)}_{x}}{{\eta }_{i}},\end{eqnarray}$to simplify the symmetry constraint condition (57) as$\begin{eqnarray}u=\displaystyle \sum _{m=1}^{N}{a}_{m}{g}_{m},\end{eqnarray}$and then to transform equations (58)–(59) into a nonlinear system of N-component ordinary differential equations in the form of$\begin{eqnarray}\begin{array}{l}({w}_{{ix}}^{2}-{\eta }_{i}^{2}{w}_{i}^{2}){\left(\displaystyle \sum _{m=1}^{N}{a}_{m}{w}_{m}\right)}^{2}\\ \ \ -\,{\left({w}_{{ixx}}-{\eta }_{i}^{2}{w}_{i}\right)}^{2}=0,\quad i=1,2,...,N,\end{array}\end{eqnarray}$with ${w}_{i}={g}_{{ix}}$. It is noted that the above system is just the one obtained in [11], which demonstrates that from different original systems and nonlocal symmetries, the same nonlinear integrable system might be established via the symmetry constraint approach.
Second, we introduce the second nonlocal symmetry (13) on the x-part of the BT, namely$\begin{eqnarray}\begin{array}{l}{u}_{x}=\displaystyle \sum _{i=1}^{N}{a}_{i}{{\rm{e}}}^{{\eta }_{i}\displaystyle \int \cos \left(\tfrac{u}{2}-\tfrac{{u}_{i}}{2}\right){\rm{d}}x}\\ \ \ \displaystyle \int {{\rm{e}}}^{-{\eta }_{i}\displaystyle \int \cos \left(\tfrac{u}{2}-\tfrac{{u}_{i}}{2}\right){\rm{d}}x}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right){\rm{d}}x,\end{array}\end{eqnarray}$then equation (59) with the above constraint (63) constitutes the second kind of the finite dimensional $(N+1)$-component integro-differential equations. Under the same introduction (60), equation (63) can be rewritten as$\begin{eqnarray}{u}_{x}=\displaystyle \sum _{m=1}^{N}{a}_{m}{g}_{{mx}}{H}_{m},\end{eqnarray}$with$\begin{eqnarray}{H}_{m}\equiv {H}_{m}(x,t)=\int {g}_{{mx}}^{-1}\sqrt{1-{\left(\displaystyle \frac{{\left(\mathrm{ln}{g}_{{mx}}\right)}_{x}}{{\eta }_{m}}\right)}^{2}}{\rm{d}}x.\end{eqnarray}$Consequently, we have a nonlinear system of N-component integro-differential equations$\begin{eqnarray}\begin{array}{l}({g}_{{ixx}}^{2}-{\eta }_{i}^{2}{g}_{{ix}}^{2}){H}_{m}{\left(\displaystyle \sum _{m=1}^{N}{a}_{m}{g}_{{mx}}\right)}^{2}\\ \ \ -\,{\left({g}_{{ixxx}}-{\eta }_{i}^{2}{g}_{{ix}}\right)}^{2}=0,\quad i=1,2,...,N,\end{array}\end{eqnarray}$with H given by (65).
Following the same way, other nonlocal symmetries can also be used to not only the x-part of the BT, but also to the t-part of the BT (55) to form new nonlinear integrable systems, but the results seem much more complicated.
Case 2: Higher dimensional nonlinear integrable systems. It is known that infinite dimensional nonlinear models can also be constructed in a similar way by introducing internal parameters, namely, imposing some internal parameter dependent symmetry constraints on the BT.
Let us take$\begin{eqnarray}{u}_{y}=\displaystyle \sum _{i=1}^{N}{a}_{i}\exp \left({\eta }_{i}\int \cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right){\rm{d}}x\right),\end{eqnarray}$as a new symmetry constraint condition, which is feasible because the mKdV-SG equation is invariant under the inner parameter y translation, and apply it on the x-part of the BT (2) to form a (1+1)-dimensional ($N+1$)-component integro-differential system$\begin{eqnarray}{u}_{y}=\displaystyle \sum _{i=1}^{N}{a}_{i}\exp \left({\eta }_{i}\int \cos \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right){\rm{d}}x\right),\end{eqnarray}$$\begin{eqnarray}{u}_{x}+{u}_{{ix}}=2{\eta }_{i}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right),\quad i=1,2,...,N,\end{eqnarray}$with arbitrary constants ai and ${\eta }_{i}$. The further application of the following transformation$\begin{eqnarray}u={u}_{i}+2\arccos \left(\displaystyle \frac{1}{{\eta }_{i}}{\left(\mathrm{ln}{f}_{{iy}}\right)}_{x}\right)=\displaystyle \sum _{m=1}^{N}{a}_{m}{f}_{m},\end{eqnarray}$on equations (68)–(69) arrives at a system of (1+1)-dimensional N-component differential equations$\begin{eqnarray}\begin{array}{l}({f}_{{ixy}}^{2}-{\eta }^{2}{f}_{{iy}}^{2}){\left(\displaystyle \sum _{m=1}^{N}{a}_{m}{f}_{{mx}}\right)}^{2}\\ \ \ -\,{\left({f}_{{ixxy}}-{\eta }_{i}^{2}{f}_{{iy}}\right)}^{2}=0.\end{array}\end{eqnarray}$
It is noted that the above system (71) is equivalent to the one presented in [11]. In addition, the symmetry condition (67) and the others with other nonlocal symmetries can also be imposed on the t-part of the BT (55) with (56) to form new higher dimensional systems. For instance, applying the following nonlocal symmetry constraint$\begin{eqnarray}\begin{array}{rcl}{u}_{y} & = & \displaystyle \sum _{i=1}^{N}{a}_{i}{{\rm{e}}}^{{\eta }_{i}\displaystyle \int \cos \left(\tfrac{u}{2}-\tfrac{{u}_{i}}{2}\right){\rm{d}}x}\\ & & \times \,\displaystyle \int {{\rm{e}}}^{-{\eta }_{i}\displaystyle \int \cos \left(\tfrac{u}{2}-\tfrac{{u}_{i}}{2}\right){\rm{d}}x}\sin \left(\displaystyle \frac{u}{2}-\displaystyle \frac{{u}_{i}}{2}\right){\rm{d}}x,\end{array}\end{eqnarray}$on the t-part of the BT (55) will lead to a (2+1)-dimensional N-component integro-differential equations, which are not given explicitly here for their complicated expressions.
5. Summary and discussions
In summary, the nonlocal symmetries and nonlocal conservation laws of the mKdV-SG equation are studied in detail. It is shown that the linearized equations of the mKdV-SG equation and its BT can give not only new nonlocal symmetries related to the BT, but also new nonlocal conservation laws related to the new nonlocal symmetries. In detail, three special nonlocal symmetries and one special conservation law are given explicitly. Then using the parameter expansion method, infinitely many nonlocal symmetries and infinitely many nonlocal conservation laws are constructed explicitly and straightforwardly. Finally, imposing symmetry constraints with the new nonlocal symmetries on the BT, finite and infinite dimensional systems of N coupled nonlinear equations are constructed, whose integrable properties need further considerations. As the mKdV-SG equation plays an important role in physics, it is really hoped that the results presented above might also find their applications in various physics.
Acknowledgments
The authors acknowledge the financial support by the National Natural Science Foundation of China (Grant Nos. 11675055 and 12071302) and the Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).
BlumanG W, ReidG J, KumeiS1988 New classes of symmetries for partial differential equations J. Math. Phys. 29 806811 DOI:10.1063/1.527974 [Cited within: 1]
BlumanG W, YüzbasiA K2020 How symmetries yield non-invertible mappings of linear partial differential equations J. Math. Anal. Appl. 491 124354 DOI:10.1016/j.jmaa.2020.124354 [Cited within: 1]
OpanasenkoS, BihloA, PopovychR O, SergyeyevA2020 Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model Physica D 411 132546 DOI:10.1016/j.physd.2020.132546 [Cited within: 1]
AliM R2019 A truncation method for solving the time-fractional Benjamin-Ono equation J. Appl. Math. 2019 3456848 DOI:10.1155/2019/3456848 [Cited within: 1]
MaW X, AliM R, SadatR2020 Analytical solutions for nonlinear dispersive physical model Complexity 2020 3714832 DOI:10.1155/2020/3714832 [Cited within: 1]
GaoX N, LouS Y, TangX Y2013 Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation J. High Energy Phys. 05 029 DOI:10.1007/JHEP05(2013)029
HuX R, LouS Y, ChenY2012 Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation Phys. Rev. E 85 056607 DOI:10.1103/PhysRevE.85.056607
WangJ Y, TangX Y, LiangZ F, LouS Y2015 Infinitely many nonlocal symmetries and conservation laws for the (1+1)-dimensional Sine-Gordon equation J. Math. Anal. Appl. 421 685696 DOI:10.1016/j.jmaa.2014.07.040 [Cited within: 3]
XinX P, LiuY T, LiuX Q2016 Nonlocal symmetries, exact solutions and conservation laws of the coupled Hirota equations Appl. Math. Lett. 55 6371 DOI:10.1016/j.aml.2015.11.009
HerederoR H, ReyesE G2012 Geometric Integrability of the Camassa-Holm Equation. II International Mathematics Research Notices 2012 30893125 DOI:10.1093/imrn/rnr120 [Cited within: 1]
ZhaoZ L, HanB2018 Residual symmetry, Bäcklund transformation and CRE solvability of a (2.1)-dimensional nonlinear system Nonlinear Dyn. 94 461474 DOI:10.1007/s11071-018-4371-2 [Cited within: 1]
PuJ C, ChenY2020 Nonlocal symmetries, Bäcklund transformation and interaction solutions for the integrable boussinesq equation Modern Phys. Let. B 34 2050288 DOI:10.1142/S0217984920502887 [Cited within: 1]
KonnoK, KameyamaW, SanukiH1974 Effect of weak dislocation potential on nonlinear wave propagation in anharmonic crystal J. Phys. Soc. Jpn. 37 171176 DOI:10.1143/JPSJ.37.171 [Cited within: 1]
YongX L, WangH, GaoJ W2014 Integrability and exact solutions of the nonautonomous mixed mKdV-sinh-Gordon equation Commun Nonlinear Sci. Numer. Simulat. 19 22342244 DOI:10.1016/j.cnsns.2013.11.011 [Cited within: 1]
ChenD Y, ZhangD J, DengS F2002 The novel multi-soliton solutions of the MKdV-sine gordon equations J. Phys. Soc. Jpn. 71 658659 DOI:10.1143/JPSJ.71.658 [Cited within: 1]
PopovS P2015 Scattering of solitons by dislocations in the modified korteweg de vries-sine-gordon equation Comput. Math. Math. Phys. 55 20142024 DOI:10.1134/S0965542515120143
PopovS P2016 Nonautonomous soliton solutions of the modified korteweg-de vries-sine-gordon equation Comput. Math. Math. Phys. 56 19291937 DOI:10.1134/S0965542516110105
WazwazA M2014 N-soliton solutions for the integrable modified KdV-sine-gordon equation Phys. Scr. 89 065805 DOI:10.1088/0031-8949/89/6/065805
WazwazA M2006 Travelling wave solutions for the MKdV-sine-Gordon and the MKdV-sinh-Gordon equations by using a variable separated ODE method Appl. Math. Comput. 181 17131719 DOI:10.1016/j.amc.2006.03.024
WazwazA M, KaurL2019 Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV-Sine-Gordon equation Nonlinear Dyn. 95 22092215 DOI:10.1007/s11071-018-4686-z [Cited within: 1]
SayedS M2013 The bäcklund transformations, exact solutions, and conservation laws for the compound modified korteweg-de vries-sine-gordon equations which describe pseudospherical surfaces J. Appl. Math. 2013 613065 DOI:10.1155/2013/613065 [Cited within: 1]
LinJ, JinX W, GaoX L, LouS Y2018 Solitons on a Periodic Wave Background of the Modified KdV-Sine-Gordon Equation Commun. Theor. Phys. 70 119126 DOI:10.1088/0253-6102/70/2/119 [Cited within: 2]
,2, Wei Ding31
