Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of t...Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.展开更多
In this paper,we investigate a(3+1)-dimensional generalized variable-coefficient shallow water wave equation,which can be used to describe the flow below a pressure surface in oceanography and atmospheric science.Empl...In this paper,we investigate a(3+1)-dimensional generalized variable-coefficient shallow water wave equation,which can be used to describe the flow below a pressure surface in oceanography and atmospheric science.Employing the Kadomtsev−Petviashvili hierarchy reduction,we obtain the semi-rational solutions which describe the lumps and rogue waves interacting with the kink solitons.We find that the lump appears from one kink soliton and fuses into the other on the x−y and x−t planes.However,on the x−z plane,the localized waves in the middle of the parallel kink solitons are in two forms:lumps and line rogue waves.The effects of the variable coefficients on the two forms are discussed.The dispersion coefficient influences the speed of solitons,while the background coefficient influences the background’s height.展开更多
In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries....In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended <em>F</em>-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.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
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the...The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.展开更多
We have utilized three novel methods,called generalized direct algebraic,modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations an...We have utilized three novel methods,called generalized direct algebraic,modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation.After substituting particular values of the parameters,different solitary wave solutions are derived from the exact traveling wave solutions.It is shown that these employed methods are more powerful tools for nonlinear wave equations.展开更多
In this paper, the (2+l)-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, ...In this paper, the (2+l)-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the (2+ l)-dimensional generalization of shallow water wave equation possesses the Palnlev6 property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation (DT) is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.展开更多
With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method ...With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.展开更多
In this article,the two variable(G'G,1/G)-expansion method is suggested to investigate new and further general multiple exact wave solutions to the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)equation and the shallow wat...In this article,the two variable(G'G,1/G)-expansion method is suggested to investigate new and further general multiple exact wave solutions to the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)equation and the shallow water wave equation which arise in mathematical physics with the aid of computer algebra software,like Mathematica.Three functions and the rational functions solution are found.The method demonstrates power,reliability and efficiency.Indeed,the method is the generalization of the well-known(G/G)-expansion method established by Wang et al.and the method also presents a wider applicability for conducting nonlinear wave equations.展开更多
The propagation of waves in dispersive media,liquid flow containing gas bubbles,fluid flow in elastic tubes,oceans and gravity waves in a smaller domain,spatio-temporal rescaling of the nonlinear wave motion are delin...The propagation of waves in dispersive media,liquid flow containing gas bubbles,fluid flow in elastic tubes,oceans and gravity waves in a smaller domain,spatio-temporal rescaling of the nonlinear wave motion are delineated by the compound Korteweg-de Vries(KdV)-Burgers equation,the(2+1)-dimensional Maccari system and the generalized shallow water wave equation.In this work,we effectively derive abundant closed form wave solutions of these equations by using the double(G′/G,1/G)-expansion method.The obtained solutions include singular kink shaped soliton solutions,periodic solution,singular periodic solution,single soliton and other solutions as well.We show that the double(G′/G,1/G)-expansion method is an efficient and powerful method to examine nonlinear evolution equations(NLEEs)in mathematical physics and scientific application.展开更多
In this paper, three types of nonlinear evolution equations such as (2n + 1)th order KdV equation, etc, are studied. And their solitary wave solutions of rational form obtained are available and possess the simplest f...In this paper, three types of nonlinear evolution equations such as (2n + 1)th order KdV equation, etc, are studied. And their solitary wave solutions of rational form obtained are available and possess the simplest form so far. At last, the Hamiltonian form of (2n + 1)th order general KdV equation is generalized.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11975156 and 12175148)。
文摘Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.
基金financially supported by the Fundamental Research Funds for the Central Universities(Grant No.BLX201927)China Postdoctoral Science Foundation(Grant No.2019M660491)the Natural Science Foundation of Hebei Province(Grant No.A2021502003).
文摘In this paper,we investigate a(3+1)-dimensional generalized variable-coefficient shallow water wave equation,which can be used to describe the flow below a pressure surface in oceanography and atmospheric science.Employing the Kadomtsev−Petviashvili hierarchy reduction,we obtain the semi-rational solutions which describe the lumps and rogue waves interacting with the kink solitons.We find that the lump appears from one kink soliton and fuses into the other on the x−y and x−t planes.However,on the x−z plane,the localized waves in the middle of the parallel kink solitons are in two forms:lumps and line rogue waves.The effects of the variable coefficients on the two forms are discussed.The dispersion coefficient influences the speed of solitons,while the background coefficient influences the background’s height.
文摘In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended <em>F</em>-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.
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The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
文摘We have utilized three novel methods,called generalized direct algebraic,modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation.After substituting particular values of the parameters,different solitary wave solutions are derived from the exact traveling wave solutions.It is shown that these employed methods are more powerful tools for nonlinear wave equations.
基金Supported by the National Natural Science Foundation of China under Grant No.61072145the Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No.SQKM201211232016
文摘In this paper, the (2+l)-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the (2+ l)-dimensional generalization of shallow water wave equation possesses the Palnlev6 property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation (DT) is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.
基金Supported by Shandong Provincial Key Laboratory of Marine Ecology and Environment&Disaster Prevention and Mitigation project under Grant No.2012010National Natural Science Foundation of China under Grant No.11271007+1 种基金Special Funds for Theoretical Physics of the National Natural Science Foundation of China under Grant No.11447205Shandong University of Science and Technology Research Fund under Grant No.2012KYTD105
文摘With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.
文摘In this article,the two variable(G'G,1/G)-expansion method is suggested to investigate new and further general multiple exact wave solutions to the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)equation and the shallow water wave equation which arise in mathematical physics with the aid of computer algebra software,like Mathematica.Three functions and the rational functions solution are found.The method demonstrates power,reliability and efficiency.Indeed,the method is the generalization of the well-known(G/G)-expansion method established by Wang et al.and the method also presents a wider applicability for conducting nonlinear wave equations.
文摘The propagation of waves in dispersive media,liquid flow containing gas bubbles,fluid flow in elastic tubes,oceans and gravity waves in a smaller domain,spatio-temporal rescaling of the nonlinear wave motion are delineated by the compound Korteweg-de Vries(KdV)-Burgers equation,the(2+1)-dimensional Maccari system and the generalized shallow water wave equation.In this work,we effectively derive abundant closed form wave solutions of these equations by using the double(G′/G,1/G)-expansion method.The obtained solutions include singular kink shaped soliton solutions,periodic solution,singular periodic solution,single soliton and other solutions as well.We show that the double(G′/G,1/G)-expansion method is an efficient and powerful method to examine nonlinear evolution equations(NLEEs)in mathematical physics and scientific application.
文摘In this paper, three types of nonlinear evolution equations such as (2n + 1)th order KdV equation, etc, are studied. And their solitary wave solutions of rational form obtained are available and possess the simplest form so far. At last, the Hamiltonian form of (2n + 1)th order general KdV equation is generalized.