By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for general...By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.展开更多
In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a ...In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.展开更多
Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutio...Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.展开更多
We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion ...We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.展开更多
One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some un...One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.展开更多
We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by v...We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrodinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.展开更多
With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pemp...With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.展开更多
In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete ...In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example,several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived.Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.展开更多
In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations....In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.展开更多
In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann the...In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.展开更多
By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other ...By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other type of the traveling wave solutions are derived.展开更多
The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behavi...The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.展开更多
For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by...For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.展开更多
A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contain...A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contains two arbitrary functions） got by means of multilinear variable separation approach for （2＋1）-dimensional KdV equation. Limiting cases are considered and some localized excitations are derived, such as dromion, multidromions, dromion-antidromion, multidromions-antidromions, and so on. Some solutions of the dromion-antidromion and multidromions-antidromions are periodic in one direction but localized in the other direction. The interaction properties of these solutions, which are numerically studied, reveal that some of them are nonelastic and some are completely elastic. Furthermore, these results are visualized.展开更多
This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear Bäcklund transformation, Lax pair, and mu...This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear Bäcklund transformation, Lax pair, and multi-shock wave solutions are successfully obtained. In addition, using the multidimensional Riemann theta functions, the periodic wave solutions of the Newell equation are constructed. On this basis, the asymptotic behavior of the periodic wave solution is given, which is the relationship between the periodic wave solution and the solitary wave solution.展开更多
In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:"">...In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different </span><span style="font-size:10.0pt;font-family:"">coefficient</span><span style="font-size:10.0pt;font-family:"">s</span><span style="font-size:10pt;font-family:""> <i>p</i>, <i>q</i> and <i>r</i> in the</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">elliptic equation. Then these solutions are</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">coupled into an auxiliary equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">and substituted into the (2+1)-dimensional KDV equation. As <span>a result,</span></span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">a large number of complex Jacobi elliptic function solutions are ob</span><span style="font-size:10pt;font-family:"">tained, and many of them have not been found in other documents. As</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10.0pt;font-family:""><span></span></span><span style="font-size:10pt;font-family:"">, some complex solitary solutions are also obtained correspondingly.</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.展开更多
For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits ...For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.展开更多
In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solu...In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.展开更多
In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are...In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.展开更多
In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generaliz...In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.展开更多
基金The project supported by the Natural Science Foundation of Anhui Province of China under Grant No. 01041188 and the Foundation of Classical Courses of Anhui Province
文摘By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11075055,61021004,10735030Shanghai Leading Academic Discipline Project under Grant No.B412Program for Changjiang Scholars and Innovative Research Team in University(IRT0734)
文摘In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.
基金The project supported by National Natural Science Foundation of China under Grant No.10771196the Natural Science Foundation of Zhejiang Province under Grant No.Y605044
文摘Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.
基金河南省自然科学基金，河南省教育厅自然科学基金，the Science Foundation of Henan University of Science and Technology
文摘We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.
基金National Natural Science Foundation of China under Grant No.10172056
文摘One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.
基金The project supported by the Natural Science Foundation of Eduction Committce of Henan Province of China under Grant No. 2003110003, and the Science Foundation of Henan University of Science and Technology under Grant Nos. 2004ZD002 and 2004ZY040
文摘We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrodinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.BUAA-SKLSDE-09KF-04+1 种基金Beijing University of Aeronautics and Astronautics,by the National Basic Research Program of China (973 Program) under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos.20060006024 and 200800130006,Chinese Ministry of Education
文摘With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.
文摘In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example,several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived.Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.
文摘In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771196 and 10831003)the Innovation Project of Zhejiang Province of China(Grant No.T200905)
文摘In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.
基金Supported by the Natural Science Foundation of Education Committee of Henan Province(2003110003)Supported by the Natural Science Foundation of Henan Province(0111050200)
文摘By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other type of the traveling wave solutions are derived.
基金supported by the National Natural Science Foundation of China (Grant No.11461022)。
文摘The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.
基金supported by the National Natural Science Foundation of China under Grant Nos.60772023 and 60372095the Key Project of the Ministry of Education under Grant No.106033+3 种基金the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China(973 Program)under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education of the Ministry of Education under Grant No.20060006024
文摘For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.
基金Foundation item： Supported by the National Natural Science Foundation of China（10647112, 10871040） Acknowledgement The authors are in debt to thank the helpful discussions with Prof Qin and Dr A P Deng.
文摘A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contains two arbitrary functions） got by means of multilinear variable separation approach for （2＋1）-dimensional KdV equation. Limiting cases are considered and some localized excitations are derived, such as dromion, multidromions, dromion-antidromion, multidromions-antidromions, and so on. Some solutions of the dromion-antidromion and multidromions-antidromions are periodic in one direction but localized in the other direction. The interaction properties of these solutions, which are numerically studied, reveal that some of them are nonelastic and some are completely elastic. Furthermore, these results are visualized.
文摘This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear Bäcklund transformation, Lax pair, and multi-shock wave solutions are successfully obtained. In addition, using the multidimensional Riemann theta functions, the periodic wave solutions of the Newell equation are constructed. On this basis, the asymptotic behavior of the periodic wave solution is given, which is the relationship between the periodic wave solution and the solitary wave solution.
文摘In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different </span><span style="font-size:10.0pt;font-family:"">coefficient</span><span style="font-size:10.0pt;font-family:"">s</span><span style="font-size:10pt;font-family:""> <i>p</i>, <i>q</i> and <i>r</i> in the</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">elliptic equation. Then these solutions are</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">coupled into an auxiliary equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">and substituted into the (2+1)-dimensional KDV equation. As <span>a result,</span></span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">a large number of complex Jacobi elliptic function solutions are ob</span><span style="font-size:10pt;font-family:"">tained, and many of them have not been found in other documents. As</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10.0pt;font-family:""><span></span></span><span style="font-size:10pt;font-family:"">, some complex solitary solutions are also obtained correspondingly.</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.
文摘For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.
基金Supported by National Natural Science Foundation of China(Grant Nos.11361069 and 11171115)
文摘In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.
基金Project supported by the Anhui Key Laboratory of Information Materials and Devices (Anhui University),China
文摘In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.
基金Supported by the National Natural Science Foundation of China(No.11401222)Natural Science Foundation of Guangdong Province(No.S2012040007959)+1 种基金The Fundamental Research Funds for the Central Universities(No.2014ZZ0064)Pearl River Science and Technology Nova Program of Guangzhou
文摘In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.
