The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painle...The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.展开更多
In this paper, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.
Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation i...Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.展开更多
In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series...In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.展开更多
By using a new method and Mathematica, the Backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are use...By using a new method and Mathematica, the Backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are used to obtain three families of solutions for WBK equations, on of which is the family of solitary wave solutions.展开更多
This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application...This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.展开更多
The residual symmetries of the Ablowitz-Kaup-Newell-Segur (AKNS) equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal r...The residual symmetries of the Ablowitz-Kaup-Newell-Segur (AKNS) equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.展开更多
In this paper, nonlocal symmetries defined by bilinear Baacklund transformation for bilinear potential Kd V(p Kd V)equation are obtained. By introducing an auxiliary variable which just satisfies the Schwartzian for...In this paper, nonlocal symmetries defined by bilinear Baacklund transformation for bilinear potential Kd V(p Kd V)equation are obtained. By introducing an auxiliary variable which just satisfies the Schwartzian form of Kd V(SKd V)equation, the nonlocal symmetry is localized and the Levi transformation is presented. Besides, based on three different types of nonlocal symmetries for potential Kd V equation, three sets of negative p Kd V hierarchies along with their bilinear forms are constructed. An impressive result is that the coefficients of the third type of(bilinear) negative p Kd V hierarchy(N 〉 0) are variable, which are obtained via introducing an arbitrary parameter by considering the translation invariance of the p Kd V equation.展开更多
In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized (2+1)-dimensional KdV equation with variable coefficients a...In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized (2+1)-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.展开更多
To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are pr...To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.展开更多
In this paper, the authors obtain the Backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones ca...In this paper, the authors obtain the Backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones can be constructed.展开更多
The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The Backlund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an...The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The Backlund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.展开更多
A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painl...A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.展开更多
We consider the solution of the good Boussinesq equation Utt -Uxx + Uxxxx = (U2)xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U(x, 0) = ε(μ + φ(x)), Ut(x, 0) = εψ(x), -∞ 〈 x 〈 ∞, where...We consider the solution of the good Boussinesq equation Utt -Uxx + Uxxxx = (U2)xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U(x, 0) = ε(μ + φ(x)), Ut(x, 0) = εψ(x), -∞ 〈 x 〈 ∞, where μ = 0, φ(x) and ψ(x) are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ(x) = μ+a sin kx, ψ(x) = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u(x, t; ε) and the N-th partial sum of the asymptotic series is bounded by εN+1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ |ε|t ≤ T and 0 ≤ |ε|≤ε0.展开更多
By means of a Painlevé-Baicklund transformation and a multi-linear separation-of-variable approach, abundant localized coherent excitations of a modified Broer-Kaup system are derived. There appear possible phase...By means of a Painlevé-Baicklund transformation and a multi-linear separation-of-variable approach, abundant localized coherent excitations of a modified Broer-Kaup system are derived. There appear possible phase shifts for the interactions of the (2+1)-dimensional novel localized structures, which are discussed in this paper.展开更多
We investigate the extended (2+ 1)-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund transformation, Lax pair, and Darboux covariant Lax ...We investigate the extended (2+ 1)-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.展开更多
Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an a...Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.展开更多
The novel Wronskian solutions of the KdV equation were obtained as limits of the soliton solutions in the Wronskian form. These solutions were verified by direct substitution to satisfy the bilinear derivative form of...The novel Wronskian solutions of the KdV equation were obtained as limits of the soliton solutions in the Wronskian form. These solutions were verified by direct substitution to satisfy the bilinear derivative form of the KdV equation and its Backlund transformation.展开更多
Taking the potential fifth-order MKdV equation as an example to introduce a possible way to construct invariance of a nonlinear PDE. Based on an obtained Backlund transformation of the potential fifth-order MKdV equat...Taking the potential fifth-order MKdV equation as an example to introduce a possible way to construct invariance of a nonlinear PDE. Based on an obtained Backlund transformation of the potential fifth-order MKdV equation and by solving the corresponding Ricatti form Lax pairs, an invariance of the potential fifth-order MKdV equation is digged out. Thus, just by differential and quadrature procedure,the solutions of the potential fifth-order MKdV equation can be obtained from a known solution.展开更多
With the help of the method that combines the first kind of elliptic equation with the function transformation, some kinds of new composite solutions of a kind of coupled Schr?dinger equation are constructed. First, a...With the help of the method that combines the first kind of elliptic equation with the function transformation, some kinds of new composite solutions of a kind of coupled Schr?dinger equation are constructed. First, a kind of function transformation is presented, and then the problem of solving solutions of a kind of coupled Schr?dinger equation can be changed to the problem of solving solutions of the first kind of elliptic equation. Then, with the help of the conclusions of the B?cklund transformation and so on of the first kind of elliptic equation, the new infinite sequence composite solutions of a kind of coupled Schr?dinger equation are constructed. These solutions are consisting of two-soliton solutions and two-period solutions and so on.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11775047,11775146,and 11865013)the Science and Technology Project Foundation of Zhongshan City,China(Grant No.2017B1016).
文摘The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.
文摘In this paper, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.
基金supported by the National Nature Science Foundation of China under Grant No.11871116Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11。
文摘Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.
文摘In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.
文摘By using a new method and Mathematica, the Backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are used to obtain three families of solutions for WBK equations, on of which is the family of solitary wave solutions.
基金sponsored by the National Natural Science Foundations of China(Nos.12301315,12235007,11975131)the Natural Science Foundation of Zhejiang Province(No.LQ20A010009).
文摘This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11305031,11365017,and 11305106)the Natural Science Foundation of Guangdong Province,China(Grant No.S2013010011546)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Grant No.LQ13A050001)the Science and Technology Project Foundation of Zhongshan,China(Grant Nos.2013A3FC0264 and 2013A3FC0334)the Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province,China(Grant No.Yq2013205)
文摘The residual symmetries of the Ablowitz-Kaup-Newell-Segur (AKNS) equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.
基金supported by the Natural Science Foundation of Zhejiang Province,China(Grant No.LQ13A010014)the National Natural Science Foundation of China(Grant Nos.11326164,11401528,and 11275072)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120076110024)
文摘In this paper, nonlocal symmetries defined by bilinear Baacklund transformation for bilinear potential Kd V(p Kd V)equation are obtained. By introducing an auxiliary variable which just satisfies the Schwartzian form of Kd V(SKd V)equation, the nonlocal symmetry is localized and the Levi transformation is presented. Besides, based on three different types of nonlocal symmetries for potential Kd V equation, three sets of negative p Kd V hierarchies along with their bilinear forms are constructed. An impressive result is that the coefficients of the third type of(bilinear) negative p Kd V hierarchy(N 〉 0) are variable, which are obtained via introducing an arbitrary parameter by considering the translation invariance of the p Kd V equation.
基金supported by the National Natural Science Foundation of China(Grant No.10831003)the Natural Science Foundation of Zhejiang Province,China(Grant Nos.Y6100791 and R6090109)
文摘In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized (2+1)-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.
基金supported by the National Natural Science Foundation of China(Grant No.10862003)the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China(Grant No.NJZZ07031)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2010MS0111)
文摘To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.
基金the NNSF(19971084,10101025 and 10231050)of China
文摘In this paper, the authors obtain the Backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones can be constructed.
基金Project supported by the Science Research Foundation of Zhanjiang Normal University (Grant No. L0803)
文摘The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The Backlund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.
基金supported by Chinese National Social Science Foundation(Grant Number:CNSSF:13CJY037)Research on the indemnificatory Apartment Construction Based on Residential Integration.
文摘A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.
基金supported by National Natural Science Foundation of China(10871199)
文摘We consider the solution of the good Boussinesq equation Utt -Uxx + Uxxxx = (U2)xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U(x, 0) = ε(μ + φ(x)), Ut(x, 0) = εψ(x), -∞ 〈 x 〈 ∞, where μ = 0, φ(x) and ψ(x) are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ(x) = μ+a sin kx, ψ(x) = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u(x, t; ε) and the N-th partial sum of the asymptotic series is bounded by εN+1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ |ε|t ≤ T and 0 ≤ |ε|≤ε0.
基金Project supported by the National Natural Science Foundation of China (Grant No 10272071) and the Key Assisted Academic Discipline of Zhejiang Province (Grant No 200337).
文摘By means of a Painlevé-Baicklund transformation and a multi-linear separation-of-variable approach, abundant localized coherent excitations of a modified Broer-Kaup system are derived. There appear possible phase shifts for the interactions of the (2+1)-dimensional novel localized structures, which are discussed in this paper.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11075055 and 11275072)the Innovative Research Team Program of the National Natural Science Foundation of China (Grant No. 61021004)+1 种基金the Shanghai Knowledge Service Platform for Trustworthy Internet of Things, China(Grant No. ZF1213)the National High Technology Research and Development Program of China (Grant No. 2011AA010101)
文摘We investigate the extended (2+ 1)-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.
基金Project supported by the National Natural Scinece Foundation of China(Grant Nos.11671219,11871446,12071304,and 12071451).
文摘Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.
文摘The novel Wronskian solutions of the KdV equation were obtained as limits of the soliton solutions in the Wronskian form. These solutions were verified by direct substitution to satisfy the bilinear derivative form of the KdV equation and its Backlund transformation.
文摘Taking the potential fifth-order MKdV equation as an example to introduce a possible way to construct invariance of a nonlinear PDE. Based on an obtained Backlund transformation of the potential fifth-order MKdV equation and by solving the corresponding Ricatti form Lax pairs, an invariance of the potential fifth-order MKdV equation is digged out. Thus, just by differential and quadrature procedure,the solutions of the potential fifth-order MKdV equation can be obtained from a known solution.
基金supported by the Natural Natural Science Foundation of China(Grant No.11361040)the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China(Grant No.NJZY12031)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2015MS0128).
文摘With the help of the method that combines the first kind of elliptic equation with the function transformation, some kinds of new composite solutions of a kind of coupled Schr?dinger equation are constructed. First, a kind of function transformation is presented, and then the problem of solving solutions of a kind of coupled Schr?dinger equation can be changed to the problem of solving solutions of the first kind of elliptic equation. Then, with the help of the conclusions of the B?cklund transformation and so on of the first kind of elliptic equation, the new infinite sequence composite solutions of a kind of coupled Schr?dinger equation are constructed. These solutions are consisting of two-soliton solutions and two-period solutions and so on.