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 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
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
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 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
The evolution and run-up of double solitary waves on a plane beach were studied numerically using the nonlinear shallow water equations(NSWEs) and the Godunov scheme. The numerical model was validated through compar...The evolution and run-up of double solitary waves on a plane beach were studied numerically using the nonlinear shallow water equations(NSWEs) and the Godunov scheme. The numerical model was validated through comparing the present numerical results with analytical solutions and laboratory measurements available for propagation and run-up of single solitary wave. Two successive solitary waves with equal wave heights and variable separation distance of two crests were used as the incoming wave on the open boundary at the toe of a slope beach. The run-ups of the first wave and the second wave with different separation distances were investigated. It is found that the run-up of the first wave does not change with the separation distance and the run-up of the second wave is affected slightly by the separation distance when the separation distance is gradually shortening. The ratio of the maximum run-up of the second wave to one of the first wave is related to the separation distance as well as wave height and slope. The run-ups of double solitary waves were compared with the linearly superposed results of two individual solitary-wave run-ups. The comparison reveals that linear superposition gives reasonable prediction when the separation distance is large, but it may overestimate the actual run-up when two waves are close.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.
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.
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.
基金Project supported by the National Natural Science Foundation of China(Grant No.51379123)the Natural Science Foundation of Shanghai Municipality(Grant No.11ZR1418200)the Shanghai Water Authority and the State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University(Grant No.GKZD010063)
文摘The evolution and run-up of double solitary waves on a plane beach were studied numerically using the nonlinear shallow water equations(NSWEs) and the Godunov scheme. The numerical model was validated through comparing the present numerical results with analytical solutions and laboratory measurements available for propagation and run-up of single solitary wave. Two successive solitary waves with equal wave heights and variable separation distance of two crests were used as the incoming wave on the open boundary at the toe of a slope beach. The run-ups of the first wave and the second wave with different separation distances were investigated. It is found that the run-up of the first wave does not change with the separation distance and the run-up of the second wave is affected slightly by the separation distance when the separation distance is gradually shortening. The ratio of the maximum run-up of the second wave to one of the first wave is related to the separation distance as well as wave height and slope. The run-ups of double solitary waves were compared with the linearly superposed results of two individual solitary-wave run-ups. The comparison reveals that linear superposition gives reasonable prediction when the separation distance is large, but it may overestimate the actual run-up when two waves are close.