This paper investigates a real version of a (2 + 1) dimensional nonlinear Schr?dinger equation through adoption of Painlevé test by means of which the (2 + 1) dimensional nonlinear Schr?dinger equation is studied...This paper investigates a real version of a (2 + 1) dimensional nonlinear Schr?dinger equation through adoption of Painlevé test by means of which the (2 + 1) dimensional nonlinear Schr?dinger equation is studied according to the Weiss et al. method and Kruskal’s simplification algorithms. According to Painlevé test, it is found that the number of arbitrary functions required for explaining the Cauchy-Kovalevskaya theorem exist. Finally, the associated B?cklund transformation and bilinear form is directly obtained from the Painlevé test.展开更多
This paper is to investigate the extended(2+1)-dimensional Konopelchenko-Dubrovsky equations,which can be applied to describing certain phenomena in the stratified shear flow,the internal and shallow-water waves, plas...This paper is to investigate the extended(2+1)-dimensional Konopelchenko-Dubrovsky equations,which can be applied to describing certain phenomena in the stratified shear flow,the internal and shallow-water waves, plasmas and other fields.Painleve analysis is passed through via symbolic computation.Bilinear-form equations are constructed and soliton solutions are derived.Soliton solutions and interactions are illustrated.Bilinear-form Backlund transformation and a type of solutions are obtained.展开更多
Using the standard Painlevé analysis and the perturbative method, the Painlevé test for the logarithmic branch is investigated. Nine arbitrary functions are obtained and the Baecklund transformation of the l...Using the standard Painlevé analysis and the perturbative method, the Painlevé test for the logarithmic branch is investigated. Nine arbitrary functions are obtained and the Baecklund transformation of the logarithmic branch is given. Using the new type Baecklund transformation, many exact solutions are obtained.展开更多
There is a close relationship between the Painlevéintegrability and other integrability of nonlinear evolution equation.By using the Weiss-Tabor-Carnevale(WTC)method and the symbolic computation of Maple,the Pain...There is a close relationship between the Painlevéintegrability and other integrability of nonlinear evolution equation.By using the Weiss-Tabor-Carnevale(WTC)method and the symbolic computation of Maple,the Painlevétest is used for the higher order generalized non-autonomous equation and the third order Korteweg-de Vries equation with variable coefficients.Finally the Painlevéintegrability condition of this equation is gotten.展开更多
A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbit...A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbitrariness of the manifold in Painlevé analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.展开更多
The Zakharov-Kuznetsov equation is proved to be nonintegrable by standard Painleve approach and three new types of soliton solutions are obtained by means of the nonstandard truncation of the extended Painleve analysi...The Zakharov-Kuznetsov equation is proved to be nonintegrable by standard Painleve approach and three new types of soliton solutions are obtained by means of the nonstandard truncation of the extended Painleve analysis approach.展开更多
The(2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani(KdVSKR)equation is studied by the singularity structure analysis.It is proven that it admits the Painlevéproperty.The Lie algebras which depend on t...The(2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani(KdVSKR)equation is studied by the singularity structure analysis.It is proven that it admits the Painlevéproperty.The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method.It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra.By selecting first-order polynomials in t,a finite-dimensional subalgebra of physical transformations is studied.The commutation relations of the subalgebra,which have been established by selecting the Laurent polynomials in t,are calculated.This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics.Meanwhile,the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.展开更多
For the propagation of the ultrashort pulses in an inhomogeneous multi-component nonlinear medium, a system of coupled equations is analytically studied in this paper. Painleve analysis shows that this system admits t...For the propagation of the ultrashort pulses in an inhomogeneous multi-component nonlinear medium, a system of coupled equations is analytically studied in this paper. Painleve analysis shows that this system admits the Painleve property under some constraints. By means of the Ablowitz-Kaup-Newell-Segur procedure, the Lax pair of this system is derived, and the Darboux transformation （DT） is constructed with the help of the obtained Lax pair. With symbolic computation, the soliton solutions are obtained by virtue of the DT algorithm. Figures are plotted to illustrate the dynamical features of the soliton solutions. Characteristics of the solitons propagating in an inhomogeneous multi-component nonlinear medium are discussed： （i） Propagation of one soliton and two-peak soliton; （ii） Elastic interactions of the parabolic two solitons; （iii） Overlap two head-on solitons and two head-on two-peak solitons; （v） Two （vi） Decomposition phenomenon of one soliton into two solitons. phenomenon between two solitons; （iv） Collision of different types of interactions of the three solitons; ultrashort-pulse propagation in the inhomogeneous multi-component The results might be useful in the study on the nonlinear media.展开更多
By Painleve analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term ...By Painleve analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.展开更多
We study the Painlevé property of the （1＋1）-dimensional equations arising from the symmetry reduction for the （2＋1）- dimensional ones. Firstly, we derive the similarity reduction of the （2＋1）-dimensional...We study the Painlevé property of the （1＋1）-dimensional equations arising from the symmetry reduction for the （2＋1）- dimensional ones. Firstly, we derive the similarity reduction of the （2＋1）-dimensional potential Calogero-Bogoyavlenskii- Schiff （CBS） equation and Konopelchenko-Dubrovsky （KD） equations with the optimal system of the admitted one-dimensional subalgebras. Secondly, by analyzing the reduced CBS, KD, and Burgers equations with Painlevé test, re-spectively, we find both the Painlevé integrability, and the number and location of resonance points are invariant, if the similarity variables include all of the independent variables.展开更多
In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soli...In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soliton solutions.展开更多
A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation.To investigate the integrability of the N = 1 supersymmetric Ito(sIto) equatio...A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation.To investigate the integrability of the N = 1 supersymmetric Ito(sIto) equation, a singularity structure analysis for this system is carried out.Through a detailed analysis in two cases by using Kruskal’s simplified method, the sIto system is found to pass the Painlevé test, and thus is Painlevé integrable.展开更多
A reaction diffusion system arising in the theory of superconductivity is considered and its m any kinds of analytic solutions are constructed by the Painleve′analysis and sim ilarity reduction m ethods.
A predator prey interaction of both populations was treated by the Painlevé analysis method. The analytic solutions for the related equations was obtained using the truncated Painlevé expansion.
In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We fin...In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.展开更多
This article considers time-dependent variable coefficients(2+1)and(3+1)-dimensional extended Sakovich equation.Painlevéanalysis and auto-Bäcklund transformation methods are used to examine both the consider...This article considers time-dependent variable coefficients(2+1)and(3+1)-dimensional extended Sakovich equation.Painlevéanalysis and auto-Bäcklund transformation methods are used to examine both the considered equations.Painlevéanalysis is appeared to test the integrability while an auto-Bäcklund transformation method is being presented to derive new analytic soliton solution families for both the considered equations.Two new family of exact analytical solutions are being obtained success-fully for each of the considered equations.The soliton solutions in the form of rational and exponential functions are being depicted.The results are also expressed graphically to illustrate the potential and physical behaviour of both equations.Both the considered equations have applications in ocean wave theory as they depict new solitary wave soliton solutions by 3D and 2D graphs.展开更多
In this paper, a variable-coefficient Benjarnin-Bona-Mahony-Burger （BBMB） equation arising as a math- ematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The in...In this paper, a variable-coefficient Benjarnin-Bona-Mahony-Burger （BBMB） equation arising as a math- ematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The inte- grability of such an equation is studied with Painlevd analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Further- more different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.展开更多
In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction s...In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is dimcult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus rn = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.展开更多
The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial different...The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.展开更多
This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to th...This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.展开更多
基金supported by the National Natural Science Foundation of China(grant No.11371361)the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology(2014)the Key Discipline Construction by China University of Mining and Technology(Grant No.XZD 201602).
文摘This paper investigates a real version of a (2 + 1) dimensional nonlinear Schr?dinger equation through adoption of Painlevé test by means of which the (2 + 1) dimensional nonlinear Schr?dinger equation is studied according to the Weiss et al. method and Kruskal’s simplification algorithms. According to Painlevé test, it is found that the number of arbitrary functions required for explaining the Cauchy-Kovalevskaya theorem exist. Finally, the associated B?cklund transformation and bilinear form is directly obtained from the Painlevé test.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023the Open Fund under Grant No.SKLSDE-2011KF-03+2 种基金Supported project under Grant No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and Astronauticsthe National High Technology Research and Development Program of China(863 Program) under Grant No.2009AA043303the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.200800130006,Chinese Ministry of Education
文摘This paper is to investigate the extended(2+1)-dimensional Konopelchenko-Dubrovsky equations,which can be applied to describing certain phenomena in the stratified shear flow,the internal and shallow-water waves, plasmas and other fields.Painleve analysis is passed through via symbolic computation.Bilinear-form equations are constructed and soliton solutions are derived.Soliton solutions and interactions are illustrated.Bilinear-form Backlund transformation and a type of solutions are obtained.
文摘Using the standard Painlevé analysis and the perturbative method, the Painlevé test for the logarithmic branch is investigated. Nine arbitrary functions are obtained and the Baecklund transformation of the logarithmic branch is given. Using the new type Baecklund transformation, many exact solutions are obtained.
基金Supported by the Shanxi Education Department Project(Grant No.J2020398)Key Natural Science Projects of Shanxi Energy Institute(Grant No.ZZ-2018003)。
文摘There is a close relationship between the Painlevéintegrability and other integrability of nonlinear evolution equation.By using the Weiss-Tabor-Carnevale(WTC)method and the symbolic computation of Maple,the Painlevétest is used for the higher order generalized non-autonomous equation and the third order Korteweg-de Vries equation with variable coefficients.Finally the Painlevéintegrability condition of this equation is gotten.
文摘A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbitrariness of the manifold in Painlevé analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.
基金supported by the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers under Grant No.355876
文摘The Zakharov-Kuznetsov equation is proved to be nonintegrable by standard Painleve approach and three new types of soliton solutions are obtained by means of the nonstandard truncation of the extended Painleve analysis approach.
基金supported by the National Natural Science Foundation of China Grant Nos.11775146,11835011 and 12105243.
文摘The(2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani(KdVSKR)equation is studied by the singularity structure analysis.It is proven that it admits the Painlevéproperty.The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method.It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra.By selecting first-order polynomials in t,a finite-dimensional subalgebra of physical transformations is studied.The commutation relations of the subalgebra,which have been established by selecting the Laurent polynomials in t,are calculated.This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics.Meanwhile,the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023by the Open Fund No.BUAA-SKLSDE-09KF-04+2 种基金Supported Project No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and Astronauticsby the National Basic Research Program of China (973 Program) under Grant No.2005CB321901by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.200800130006,Chinese Ministry of Education
文摘For the propagation of the ultrashort pulses in an inhomogeneous multi-component nonlinear medium, a system of coupled equations is analytically studied in this paper. Painleve analysis shows that this system admits the Painleve property under some constraints. By means of the Ablowitz-Kaup-Newell-Segur procedure, the Lax pair of this system is derived, and the Darboux transformation （DT） is constructed with the help of the obtained Lax pair. With symbolic computation, the soliton solutions are obtained by virtue of the DT algorithm. Figures are plotted to illustrate the dynamical features of the soliton solutions. Characteristics of the solitons propagating in an inhomogeneous multi-component nonlinear medium are discussed： （i） Propagation of one soliton and two-peak soliton; （ii） Elastic interactions of the parabolic two solitons; （iii） Overlap two head-on solitons and two head-on two-peak solitons; （v） Two （vi） Decomposition phenomenon of one soliton into two solitons. phenomenon between two solitons; （iv） Collision of different types of interactions of the three solitons; ultrashort-pulse propagation in the inhomogeneous multi-component The results might be useful in the study on the nonlinear media.
文摘By Painleve analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.
基金Project supported by the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2011AQ017 and ZR2010AM028)the Fundamental Research Funds for the Central Universities of Ministry of Education of China(Grant No.13CX02010A)
文摘We study the Painlevé property of the （1＋1）-dimensional equations arising from the symmetry reduction for the （2＋1）- dimensional ones. Firstly, we derive the similarity reduction of the （2＋1）-dimensional potential Calogero-Bogoyavlenskii- Schiff （CBS） equation and Konopelchenko-Dubrovsky （KD） equations with the optimal system of the admitted one-dimensional subalgebras. Secondly, by analyzing the reduced CBS, KD, and Burgers equations with Painlevé test, re-spectively, we find both the Painlevé integrability, and the number and location of resonance points are invariant, if the similarity variables include all of the independent variables.
基金Project supported by the National Natural Science Foundation of China (Grant No 10671206)
文摘In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soliton solutions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975156 and 11775146)the Natural Science Foundation of Zhejiang Province,China(Grant No.LY18A050001)
文摘A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation.To investigate the integrability of the N = 1 supersymmetric Ito(sIto) equation, a singularity structure analysis for this system is carried out.Through a detailed analysis in two cases by using Kruskal’s simplified method, the sIto system is found to pass the Painlevé test, and thus is Painlevé integrable.
文摘A reaction diffusion system arising in the theory of superconductivity is considered and its m any kinds of analytic solutions are constructed by the Painleve′analysis and sim ilarity reduction m ethods.
文摘A predator prey interaction of both populations was treated by the Painlevé analysis method. The analytic solutions for the related equations was obtained using the truncated Painlevé expansion.
基金the National Natural Science Foundation of China under Grant No.11772017the Fundamental Research Funds for the Central Universities
文摘In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.
文摘This article considers time-dependent variable coefficients(2+1)and(3+1)-dimensional extended Sakovich equation.Painlevéanalysis and auto-Bäcklund transformation methods are used to examine both the considered equations.Painlevéanalysis is appeared to test the integrability while an auto-Bäcklund transformation method is being presented to derive new analytic soliton solution families for both the considered equations.Two new family of exact analytical solutions are being obtained success-fully for each of the considered equations.The soliton solutions in the form of rational and exponential functions are being depicted.The results are also expressed graphically to illustrate the potential and physical behaviour of both equations.Both the considered equations have applications in ocean wave theory as they depict new solitary wave soliton solutions by 3D and 2D graphs.
文摘In this paper, a variable-coefficient Benjarnin-Bona-Mahony-Burger （BBMB） equation arising as a math- ematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The inte- grability of such an equation is studied with Painlevd analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Further- more different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271211,11275072,11435005K.C.Wong Magna Fund in Ningbo University
文摘In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is dimcult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus rn = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.
基金supported by the National Natural Science meters restrain as the relation:Foundation of China Grant No.11775146.
文摘The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.
基金Supported by National Natural Science Foundation of China under Grant Nos.11371293,11505090the Natural Science Foundation of Shaanxi Province under Grant No.2014JM2-1009+1 种基金Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009the Science and Technology Innovation Foundation of Xi’an under Grant No.CYX1531WL41
文摘This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.
