By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple...By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple manner.展开更多
In this paper, two transformations are introduced to solve sinh-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in o...In this paper, two transformations are introduced to solve sinh-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the sinh-Gordon equation.展开更多
In this paper,the complex multi-symplectic method and the implementation of the generalized sinhGordon equation are investigated in detail.The multi-symplectic formulations of the generalized sinh-Gordon equation in H...In this paper,the complex multi-symplectic method and the implementation of the generalized sinhGordon equation are investigated in detail.The multi-symplectic formulations of the generalized sinh-Gordon equation in Hamiltonian space are presented firstly.The complex method is introduced and a complex semi-implicit scheme with several discrete conservation laws(including a multi-symplectic conservation law(CLS),a local energy conservation law(ECL) as well as a local momentum conservation law(MCL)) is constructed to solve the partial differential equations(PDEs) that are derived from the generalized sinh-Gordon equation numerically.The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior and high accuracy.展开更多
The modified tanh-coth function method is used to obtain new exact travelling wave solutions for Zhiber-Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd-Bullough-Mikhailov equa...The modified tanh-coth function method is used to obtain new exact travelling wave solutions for Zhiber-Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd-Bullough-Mikhailov equation, and Tzitzeica-Dodd-Bullough equation. Exact travelling wave solutions for each equation are derived and expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The modified tanh-coth function method is easy to execute, brief, efficient, and can be used to solve many other nonlinear evolution equations.展开更多
The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konop...The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.展开更多
The existence of a global periodic attractor for a sinh-Gordon equation with the homogeneous Dirichlet boundary condition, nonlinear damping and periodic forcing is proved under some condition on the parameters.
We show that the scattering operator carries a band in H^s (R^n)×H^(s-1)(R^n) into H^s (R^n)× H^(s-1)(R^n) for the sinh-Gordon equation and an analogous result also holds true for the nonlinear Schrdinger eq...We show that the scattering operator carries a band in H^s (R^n)×H^(s-1)(R^n) into H^s (R^n)× H^(s-1)(R^n) for the sinh-Gordon equation and an analogous result also holds true for the nonlinear Schrdinger equation with an exponential nonlinearity, where s≥n/2 is arbitrary and n≥2. Therefore, the scattering operators are infinitely smooth for the above two equations.展开更多
In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarch...In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.展开更多
We investigate the bounded travelling wave solutions of the Biswas-Arshed model(BAM)including the low group velocity dispersion and excluding the self-phase modulation.We integrate the nonlinear structure of the model...We investigate the bounded travelling wave solutions of the Biswas-Arshed model(BAM)including the low group velocity dispersion and excluding the self-phase modulation.We integrate the nonlinear structure of the model to obtain bounded optical solitons which pass through the optical fibers in the non-Kerr media.The bifurcation technique of the dynamical system is used to achieve the parameter bifurcation sets and split the parameter space into various areaswhich correspond to different phase portraits.All bounded optical solitons and bounded periodic wave solutions are identified and derived conforming to each region of these phase portraits.We also apply the extended sinh-Gordon equation expansion and the generalized Kudryashov integral schemes to obtain additional bounded optical soliton solutions of the BAM nonlinearity.We present more bounded optical shock waves,the bright-dark solitary wave,and optical rogue waves for the structure model via these schemes in different aspects.展开更多
This paper presents the application of a new method for obtaining new exact solutions of some well-known nonlinear partial differential equations.The Rational Hyperbolic method is used for handling the Zhiber-Shabat e...This paper presents the application of a new method for obtaining new exact solutions of some well-known nonlinear partial differential equations.The Rational Hyperbolic method is used for handling the Zhiber-Shabat equation and the related equations such as Liouville,Sinh-Gordon,Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations.We show power of the Rational Hyperbolic method that is simple and effective for solving nonlinear partial differential equations.展开更多
By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact sol...By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.展开更多
基金supported by the National Natural Science Foundation of China (Grant No 10672053) the Natural Science Foundation of Hunan Province of China (Grant No 05JJ30007)the Scientific Research Fund of Hunan Institute of Science and Technology of China (Grant No 2007Y047)
文摘By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple manner.
基金The project supported by National Natural Science Foundation of China under Grant No. 40305006
文摘In this paper, two transformations are introduced to solve sinh-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the sinh-Gordon equation.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10572119,10772147,and 10632030)the Doctoral Program Foundation of Education Ministry of China (Grant No.20070699028)+2 种基金the China Postdoctoral Science Foundation (Grant No. 20090450170)the National Natural Science Foundation of Shaanxi Province of China (Grant No.2006A07)the NPU Foundation for Fundamental Research and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (Grant No.GZ0802)
文摘In this paper,the complex multi-symplectic method and the implementation of the generalized sinhGordon equation are investigated in detail.The multi-symplectic formulations of the generalized sinh-Gordon equation in Hamiltonian space are presented firstly.The complex method is introduced and a complex semi-implicit scheme with several discrete conservation laws(including a multi-symplectic conservation law(CLS),a local energy conservation law(ECL) as well as a local momentum conservation law(MCL)) is constructed to solve the partial differential equations(PDEs) that are derived from the generalized sinh-Gordon equation numerically.The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior and high accuracy.
文摘The modified tanh-coth function method is used to obtain new exact travelling wave solutions for Zhiber-Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd-Bullough-Mikhailov equation, and Tzitzeica-Dodd-Bullough equation. Exact travelling wave solutions for each equation are derived and expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The modified tanh-coth function method is easy to execute, brief, efficient, and can be used to solve many other nonlinear evolution equations.
基金supported by the National Natural Science Foundation of China under Grant No.10672053the Scientific Research Fund of the Education Department of Hunan Province under Grant No.07D064
文摘The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.
基金supported by the National Natural Science Foundation of China.
文摘The existence of a global periodic attractor for a sinh-Gordon equation with the homogeneous Dirichlet boundary condition, nonlinear damping and periodic forcing is proved under some condition on the parameters.
基金Supported by the National Natural Science Foundation of China. Grant 19901007.
文摘We show that the scattering operator carries a band in H^s (R^n)×H^(s-1)(R^n) into H^s (R^n)× H^(s-1)(R^n) for the sinh-Gordon equation and an analogous result also holds true for the nonlinear Schrdinger equation with an exponential nonlinearity, where s≥n/2 is arbitrary and n≥2. Therefore, the scattering operators are infinitely smooth for the above two equations.
基金supported by National Basic Research Program of China (973 Program) (Grant No. 2007CB814800)National Natural Science Foundation of China (Grant Nos. 10801083, 10901090)+1 种基金Chinese Universities Scientific Fund (Grant No. 2009-2-05)the Basic Research Fund of China Agricultural University (Grant No. 2007036)
文摘In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.
基金supported by the Deanship of ScientificResearch,Prince Sattam bin Abdulaziz University,Alkharj,Saudi Arabia,under Grant No.2021/01/19122.
文摘We investigate the bounded travelling wave solutions of the Biswas-Arshed model(BAM)including the low group velocity dispersion and excluding the self-phase modulation.We integrate the nonlinear structure of the model to obtain bounded optical solitons which pass through the optical fibers in the non-Kerr media.The bifurcation technique of the dynamical system is used to achieve the parameter bifurcation sets and split the parameter space into various areaswhich correspond to different phase portraits.All bounded optical solitons and bounded periodic wave solutions are identified and derived conforming to each region of these phase portraits.We also apply the extended sinh-Gordon equation expansion and the generalized Kudryashov integral schemes to obtain additional bounded optical soliton solutions of the BAM nonlinearity.We present more bounded optical shock waves,the bright-dark solitary wave,and optical rogue waves for the structure model via these schemes in different aspects.
文摘This paper presents the application of a new method for obtaining new exact solutions of some well-known nonlinear partial differential equations.The Rational Hyperbolic method is used for handling the Zhiber-Shabat equation and the related equations such as Liouville,Sinh-Gordon,Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations.We show power of the Rational Hyperbolic method that is simple and effective for solving nonlinear partial differential equations.
基金Project supported by the National Natural Science Foundation of China (Grant No 10672053)
文摘By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.
