We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.
By means of the modified Clarkson and Kruskal （CK） direct method and the variable separation approach, we investigate the （24-1）-dimensional Ito equation which was constructed by Ito in 1980. The full symmetry group with the Kac-Moody-Virasoro algebra structure and the variable separation solutions are obtained. By selecting appropriate arbitrary functions, some special soliton excitations are shown graphically. The results presented here would be beneficial for understanding the （2-t-1）-dimensional Ito equation better.
In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the（2＋1）-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion（CRE） solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.
In this paper, nonlocal symmetries defined by bilinear Baacklund transformation for bilinear potential Kd V（p Kd V）equation are obtained. By introducing an auxiliary variable which just satisfies the Schwartzian form of Kd V（SKd V）equation, the nonlocal symmetry is localized and the Levi transformation is presented. Besides, based on three different types of nonlocal symmetries for potential Kd V equation, three sets of negative p Kd V hierarchies along with their bilinear forms are constructed. An impressive result is that the coefficients of the third type of（bilinear） negative p Kd V hierarchy（N 〉 0） are variable, which are obtained via introducing an arbitrary parameter by considering the translation invariance of the p Kd V equation.
In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equanons is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the （2＋1）-dimensional Burgers equation are presented.
We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.
The relation between the Camassa–Holm equation and the Olver–Rosenau–Qiao equation is obtained,and we connect a new Camassa–Holm type equation proposed by Qiao etc. with the first negative flow of the Kd V hierarchy by a reciprocal transformation.
The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev~ equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.
