Residual Symmetry of the Alice-Bob Modified Korteweg-de Vries Equation

Starting from the truncated Painlev′e expansion, the residual symmetry of the Alice-Bob modified Kortewegde Vries(AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transformed into an enlarged system by introducing one new variable. Based on Lie's first theorem, the finite transformation is obtained from the localized residual symmetry. Further, considering the linear superposition of multiple residual symmetries gives rises to N-th B?cklund transformation in the form of the determinant. Moreover, the P_sT_d(the shifted parity and delayed time reversal) symmetric exact solutions(including invariant solution, breaking solution and breaking interaction solution) of AB-mKdV equation are presented and two classes of interaction solutions are depicted by using the particular functions with numerical simulation.

The explicit symmetry breaking solutions of the Kadomtsev-Petviashvili equation

To describe two correlated events,the Alice-Bob(AB)systems were constructed by Lou through the symmetry of the shifted parity,time reversal and charge conjugation.In this paper,the coupled AB system of the Kadomtsev-Petviashvili equation,which is a useful model in natural science,is established.By introducing an extended Backlund transformation and its bilinear formation,the symmetry breaking soliton,lump and breather solutions of this system are derived with the aid of some ansatze functions.Figures show these fascinating symmetry breaking structures of the explicit solutions.

Breather, lump, and interaction solutions to a nonlocal KP system

A new high-dimensional two-place Alice–Bob-Kadomtsev–Petviashvili(AB-KP)equation is proposed by applying the Alice–Bob-Bob–Alice principle and shifted-parity,delayed time reversal,charge conjugation(■)principle to the usual KP equation.Based on the dependent variable transformation,the bilinear form of the AB-KP system is constructed.Explicit trigonometric-hyperbolic,rational and rational-hyperbolic solutions are presented by taking advantage of the Hirota bilinear method.The obtained breather,lump,and interaction solutions enrich the solution structure of nonlocal nonlinear systems.The three-dimensional graphs of these nonlinear wave solutions are demonstrated by choosing the specific parameters.

Lump and Stripe Soliton Solutions to the Generalized Nizhnik-Novikov-Veselov Equation

With the aid of the truncated Painlev expansion, a set of rational solutions of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov (GNNV) equation with the quadratic function which contains one lump soliton is derived. By combining this quadratic function and an exponential function, the fusion and fission phenomena occur between one lump soliton and a stripe soliton which are two kinds of typical local excitations. Furthermore, by adding a corresponding inverse exponential function to the above function, we can derive the solution with interaction between one lump soliton and a pair of stripe solitons. The dynamical behaviors of such local solutions are depicted by choosing some appropriate parameters.

The D’Alembert type waves and the soliton molecules in a (2+1)-dimensional Kadomtsev-Petviashvili with its hierarchy equation

For a one(2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D’Alembert type solution is derived first through the traveling wave transformation which contains several special kink molecule structures. Further, after introducing the B?cklund transformation and an auxiliary variable, the N-soliton solution which contains some soliton molecules for this equation, is presented through its Hirota bilinear form. The concrete molecules including line solitons, breathers and a lump as well as several interactions of their hybrid are shown with the aid of special conditions and parameters. All these dynamical features are demonstrated through the different figures.