Applying the homogeneous balance method, we have found the explicit and soliton solutions and given a successive formula of finding explicit solutions to the(2+1)-dimensional Broer-Kaup equations. Moreover, by using t...Applying the homogeneous balance method, we have found the explicit and soliton solutions and given a successive formula of finding explicit solutions to the(2+1)-dimensional Broer-Kaup equations. Moreover, by using the Lie group method, we have discussed the similarity solutions to the (2+1)-dimensional Broer-Kanp equations.展开更多
The integrability of the （2＋l）-dimensional Broer-Kaup equation with variable coefficients （VCBK） is verified by finding a transformation mapping it to the usual （2＋l）-dimensional Broer-Kaup equation （BK）. Th...The integrability of the （2＋l）-dimensional Broer-Kaup equation with variable coefficients （VCBK） is verified by finding a transformation mapping it to the usual （2＋l）-dimensional Broer-Kaup equation （BK）. Thus the solutions of the （2＋1）-dimensional VCBK are obtained by making full use of the known solutions of the usual （2＋1）dimensional IRK. Two new integrable models are given by this transformation, their dromion-like solutions and rogue wave solutions are also obtained. Further, the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the 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 improved tanh function method [Commun. Theor. Phys. （Beijing, China） 43 （2005） 585] is further improved by generalizing the ansatz solution of the considered equation. As its application, the （2＋1）-dimensio...The improved tanh function method [Commun. Theor. Phys. （Beijing, China） 43 （2005） 585] is further improved by generalizing the ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Broer-Kaup equations are considered and abundant new exact non-travelling wave solutions are obtained.展开更多
In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions...In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.展开更多
Based on a known transform, the exact solutions of (2+1)-dimensional Broer-Kaup equations are inves tigated by using the method of direct integral. A kind of new exact solutions of Broer-Kaup equations are obtained, w...Based on a known transform, the exact solutions of (2+1)-dimensional Broer-Kaup equations are inves tigated by using the method of direct integral. A kind of new exact solutions of Broer-Kaup equations are obtained, which contain previous results about solitary wave solutions.展开更多
By a known transformation, (2+1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of red...By a known transformation, (2+1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.展开更多
文摘Applying the homogeneous balance method, we have found the explicit and soliton solutions and given a successive formula of finding explicit solutions to the(2+1)-dimensional Broer-Kaup equations. Moreover, by using the Lie group method, we have discussed the similarity solutions to the (2+1)-dimensional Broer-Kanp equations.
基金Supported by the National Natural Science Foundation of China under Grant No.10971109K.C. Wong Magna Fund in Ningbo Universitythe Natural Science Foundation of Ningbo under Grant No.2011A610179
文摘The integrability of the （2＋l）-dimensional Broer-Kaup equation with variable coefficients （VCBK） is verified by finding a transformation mapping it to the usual （2＋l）-dimensional Broer-Kaup equation （BK）. Thus the solutions of the （2＋1）-dimensional VCBK are obtained by making full use of the known solutions of the usual （2＋1）dimensional IRK. Two new integrable models are given by this transformation, their dromion-like solutions and rogue wave solutions are also obtained. Further, the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.
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.
文摘The improved tanh function method [Commun. Theor. Phys. （Beijing, China） 43 （2005） 585] is further improved by generalizing the ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Broer-Kaup equations are considered and abundant new exact non-travelling wave solutions are obtained.
文摘In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.
文摘Based on a known transform, the exact solutions of (2+1)-dimensional Broer-Kaup equations are inves tigated by using the method of direct integral. A kind of new exact solutions of Broer-Kaup equations are obtained, which contain previous results about solitary wave solutions.
文摘By a known transformation, (2+1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.
