We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higheror...We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higherorder GCSs and show the main reduction procedure by some examples. The obtained reductions cannot be derived within the framework of the standard Lie approach.展开更多
The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. ...The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. Among those, a lot of solutions are new.展开更多
It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without mo...It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.展开更多
The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A comple...The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-B¨acklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.展开更多
In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut ＋ auPux ＋ bu2pux ＋ Uzzz = 0 （b 〉 0, p 〉 0）. Our results imply that orbital stability of solitary w...In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut ＋ auPux ＋ bu2pux ＋ Uzzz = 0 （b 〉 0, p 〉 0）. Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term bu2pux, but also the nonlinear term auPux. For the case of b 〉 0 and 0 〈 p ≤ 2, we obtain that the positive solitary wave Ul（X - ct） is stable when a 〉 0, while that unstable when a 〈 0. The stability for negative solitary wave u2（x - ct） is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p= 2 and a〉0.展开更多
We consider the generalized integrable fifth order nonlinear Korteweg-de Vries （fKdV） equation. The extended Tanh method has been used rigorously, by com- putational program MAPLE, for solving this fifth order nonli...We consider the generalized integrable fifth order nonlinear Korteweg-de Vries （fKdV） equation. The extended Tanh method has been used rigorously, by com- putational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.展开更多
The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be construc...The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.展开更多
We discuss the possible nonlinear waves of atomic matter waves in a cigar-shaped Bose-Einstein condensatewith dissipation. The waves can be described by a KdV-type equation. The KdV-type equation has a solitary wave s...We discuss the possible nonlinear waves of atomic matter waves in a cigar-shaped Bose-Einstein condensatewith dissipation. The waves can be described by a KdV-type equation. The KdV-type equation has a solitary wave solution. The amplitude, speed, and width of the wave vary exponentially with time t. The dissipative term of ~/ plays an important role for the wave amplitude, speed, and width. Comparisons have been given between the analytical solutions and the numerical results. It is shown that both are in good agreement.展开更多
基金supported by National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higherorder GCSs and show the main reduction procedure by some examples. The obtained reductions cannot be derived within the framework of the standard Lie approach.
基金The project supported by Scientific Reseaxch Fund of Education Department of Heilongjiang Province of China under Grant No. 11511008
文摘The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. Among those, a lot of solutions are new.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11675084 and 11435005the K.C.Wong Magna Fund in Ningbo University
文摘It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.
基金Supported by the National Science Foundation of China under Grant Nos.11371293,11501419the Mathematical Discipline Foundation of Shaanxi Province of China under Grant No.14TSXK02+1 种基金the Natural Science Foundation of Weinan Normal University under Grant No.16ZRRC05 and 15YKS005Natural Science Foundation of Hebei Province of China under Grant No.A2018207030
文摘The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-B¨acklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.
基金Supported by the National Natural Science Foundation of China(No.11471215)Innovation Program of Shanghai Municipal Education Commission(No.13ZZ118)+1 种基金Shanghai Leading Academic Discipline Project(No.XTKX2012)Hujiang Foundation of China(No.B14005)
文摘In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut ＋ auPux ＋ bu2pux ＋ Uzzz = 0 （b 〉 0, p 〉 0）. Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term bu2pux, but also the nonlinear term auPux. For the case of b 〉 0 and 0 〈 p ≤ 2, we obtain that the positive solitary wave Ul（X - ct） is stable when a 〉 0, while that unstable when a 〈 0. The stability for negative solitary wave u2（x - ct） is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p= 2 and a〉0.
文摘We consider the generalized integrable fifth order nonlinear Korteweg-de Vries （fKdV） equation. The extended Tanh method has been used rigorously, by com- putational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10675065)the Scientific Research Fundof the Education Department of Zhejiang Province of China (Grant No. 20070979)
文摘The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.
基金The project partially supported by National Natural Science Foundation of China under Grant No.10575082the Natural Science Foundation of Gansu Province under Grant No.3ZS061-A25-014the Natural Science Foundation of Northwest Normal University under Grant No.NWNU-KJCXGC-03-17
文摘We discuss the possible nonlinear waves of atomic matter waves in a cigar-shaped Bose-Einstein condensatewith dissipation. The waves can be described by a KdV-type equation. The KdV-type equation has a solitary wave solution. The amplitude, speed, and width of the wave vary exponentially with time t. The dissipative term of ~/ plays an important role for the wave amplitude, speed, and width. Comparisons have been given between the analytical solutions and the numerical results. It is shown that both are in good agreement.
