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
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
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
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, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This me...In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.展开更多
The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed aux...The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.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
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.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.
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Supported by Natural Science Foundation of Shandong Province of China under Grant No.ZR2013AQ009National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant No.201310433031Doctoral initializing Foundation of Shandong University of Technology of China under Grant No.4041-413030
文摘In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.
文摘The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.
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.
