Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters...Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.展开更多
The Whitham-Broer-Kaup model is widely used to study the tsunami waves.The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method.20 sets of solution...The Whitham-Broer-Kaup model is widely used to study the tsunami waves.The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method.20 sets of solutions are obtained of which,to the best of the authors’knowledge,some have not been reported in literature.Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.展开更多
In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is prop...In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.展开更多
In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and ...In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.展开更多
In this paper,the dispersive coupled Whitham-Broer-Kaup(DCWBK)equation with time-dependent coefficients describing the propagation of the shallow water waves are obtained.The propagation of solitons and elliptic(or ch...In this paper,the dispersive coupled Whitham-Broer-Kaup(DCWBK)equation with time-dependent coefficients describing the propagation of the shallow water waves are obtained.The propagation of solitons and elliptic(or chirped)waves can be manipulated by suitable variations of the dispersion coefficient.Here,controllable transmission of the surface waves for soliton similariton pairs with the snoidal backgrounds is considered.It is found that,when the dispersion coefficient is taken as increasing,the velocity is increasing with the dispersion coefficient increasing.While this holds vice versa for the height of propagation wave.展开更多
This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for shor...This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short.The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ),H(ξ)),and then give its global phase portraits.The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions.The authors study the dissipation effect on the solutions,find out a critical value r*,and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater,i.e.,|r| ≥ r*,while it appears as a damped oscillatory wave if the dissipation effect is smaller,i.e.,|r| < r*.Two solitary wave solutions to the WBK equation without dissipation effect is also obtained.Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits,the authors obtain all approximate damped oscillatory solutions (u(ξ),H(ξ)) under various conditions by using the undetermined coefficients method.Finally,the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.展开更多
文摘Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.
基金Supported by the National Natural Science Foundation of China(10771072)the Natural Science Foundation of Inner Mongolia(2009 MS0108)+1 种基金the High Education Science Research Programof Inner Mongolia(NJ10045)the Initial Funding of Scientific Research Project for Ph.D.of Inner Mongolia Normal University and the Natural Science Foundation of Inner Mongolia Normal University(ZRYB08017)
基金Project supported by the National Natural Science Foundation of China(No.11872241)the Discovery Early Career Researcher Award(No.DE150100169)the Centre of Excellence Grant funded by the Australian Research Council(No.CE140100003)。
文摘The Whitham-Broer-Kaup model is widely used to study the tsunami waves.The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method.20 sets of solutions are obtained of which,to the best of the authors’knowledge,some have not been reported in literature.Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.
基金Project supported by the National Natural Science Foundation of China(11501076)General Scientific Research Project of Liaoning Province(L2014279)+1 种基金Natural Science Foundation of Liaoning Province(20170540103)Foundation of Dalian Ocean University(HDYJ201409)
文摘In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.
文摘In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.
文摘In this paper,the dispersive coupled Whitham-Broer-Kaup(DCWBK)equation with time-dependent coefficients describing the propagation of the shallow water waves are obtained.The propagation of solitons and elliptic(or chirped)waves can be manipulated by suitable variations of the dispersion coefficient.Here,controllable transmission of the surface waves for soliton similariton pairs with the snoidal backgrounds is considered.It is found that,when the dispersion coefficient is taken as increasing,the velocity is increasing with the dispersion coefficient increasing.While this holds vice versa for the height of propagation wave.
基金Project supported by the National Natural Science Foundation of China (No.11071164)the Natural Science Foundation of Shanghai (No.10ZR1420800)the Shanghai Leading Academic Discipline Project (No.S30501)
文摘This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short.The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ),H(ξ)),and then give its global phase portraits.The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions.The authors study the dissipation effect on the solutions,find out a critical value r*,and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater,i.e.,|r| ≥ r*,while it appears as a damped oscillatory wave if the dissipation effect is smaller,i.e.,|r| < r*.Two solitary wave solutions to the WBK equation without dissipation effect is also obtained.Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits,the authors obtain all approximate damped oscillatory solutions (u(ξ),H(ξ)) under various conditions by using the undetermined coefficients method.Finally,the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.
