S Y Lou

On the coherent structures of the Nizhnik–Novikov–Veselov equation

Abstract A variable separation approach is used to obtain exact solutions of high-dimensional nonlinear physical models. Taking the Nizhnik–Novikov–Veselov (NNV) equation as a simple example, we show that a high-dimensional nonlinear physical model may have quite rich localized coherent structures. For the NNV equation, the richness of the localized structures caused by the entrance of two variable-separated arbitrary functions. For some special selections of the arbitrary functions, it is shown that the localized structures of the NNV equation may be dromions, lumps, breathers, instantons and ring solitons, etc.

Painlevé analysis and special solutions of generalized Broer-Kaup equations

Abstract The (1+1)-dimensional Broer–Kaup system which describes the propagation of shallow water waves is extended to a generalized (2+1)-dimensional model with Painleve property. Some special exact solutions are obtained by using the truncated Painleve expansion. The generalized (2+1)-dimensional Broer–Kaup system also possesses quite rich localized excitations as other well-known (2+1)-dimensional integrable models. Especially, the elastic (pass through) interaction property without phase shift for two ring type solitons is graphically exhibited.

Nonsingular positon and complexiton solutions for the coupled KdV system

Abstract Taking the coupled KdV system as a simple example, analytical and nonsingular complexiton solutions are firstly discovered in this Letter for integrable systems. Additionally, the analytical and nonsingular positon–negaton interaction solutions are also firstly found for S-integrable model. The new analytical positon, negaton and complexiton solutions of the coupled KdV system are given out both analytically and graphically by means of the iterative Darboux transformations.

Painlevé integrability of two sets of nonlinear evolution equations with nonlinear dispersions

Abstract It is proven that the nonlinear evolution equations ( K ( m , n ) equations), u t +( u m ) x +( u n ) xxx =0 are Painleve integrable for n = m −2 and n = m −1 with positive integer n . Especially, the solutions of the K (3,2) and K (4,2) models are single valued not only about a movable singularity manifold but also about a movable zero manifold. By using the general hodograph transformation, we know that there are five integrable K ( m , n ) models for negative n , K(− 1 2 ,− 1 2 ),K( 3 2 ,− 1 2 ),K( 1 2 ,− 1 2 ),K(−1,−2) and K (−2,−2), which are equivalent to the potential KdV, mKdV and CDF equations. However, the K ( m , n ) models for positive n are note equivalent to the known third order semilinear integrable ones.

High-dimensional Virasoro integrable models and exact solutions

Abstract In this Letter, some Virasoro integrable models are obtained by means of the realizations of the generalized centerless Virasoro type symmetry algebra, [σ(f 1 ),σ(f 2 )]=σ( f 2 f 1 − f 1 f 2 ) . Two of them are (3+1)-dimensional extensions of the Nizhnik–Novikov–Veselov equation and breaking soliton equation. Some special type of high-dimensional soliton solutions like the camber solitons, multiple ring solitons and multiple dromion solutions for the breaking soliton equation is discussed. The interaction between two ring solitons is completely elastic. Whether the method can be used to find some (3+1)-dimensional models integrable by the inverse scattering transformation remains still open.

Reflection and reconnection interactions of resonant dromions

The interactions between dromions have still not yet been understood very well. In this short paper, two new types of dromion interactions, dromion reflection and reconnection, are reported for the generalized Broer–Kaup system as a representative example. Dromion reflection is when a dromion behaves like a ball reflected by a wall. Essentially, it is a dromion interacting with an invisible ghost wall caused by a ghost line soliton. While dromion reconnection interaction is the scenario where the bounded dromions may be opened and then reconnected during the interaction. These two novel interaction phenomena are quite universal in high dimensions.

Special Bäcklund transformations and nonlinear superpositions for the non-integrable ϕ4 field model

Some special Backlund transformation (BT) theorems and a particular nonlinear superposition theorem are established to find exact solutions for a non-integrable model, the (N+1)-dimensional 4 scalar field. Some new types of exact solutions such as the conoid periodic–periodic interaction waves and the periodic–solitary wave interaction solutions are explicitly given. The interaction solutions possess abundant structures, thanks to the intrusion of some arbitrary functions in the expressions of the special solutions.

The bilinear negative Kadomtsev–Petviashvili system and its Kac–Moody–Virasoro symmetry group

The symmetry transformation group of the bilinear negative Kadomtsev– Petviashvili system is studied by means of a direct method. The Kac–Moody– Virasoro-type Lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group.

Kac-Moody-Virasoro symmetry algebra and symmetry reductions of the bilinear sinh-Gordon equation in (2 + 1)-dimensions

By means of the formal series symmetry approach proposed in [1], infinite many symmetries and the corresponding Kac–Moody–Virasoro Lie symmetry algebra of a new bilinear (2 + 1)-dimensional sinh-Gordon equation are given. Then, the obtained symmetries are used to get the symmetry reductions of the model. From one of the special reduction we know that the bilinear form of the first member of the negative Kadomtsev–Petviashvili hierarchy is not only a (2 + 1)-dimensional sinh-Gordon extension but also a novel (2 + 1)-dimensional classical Boussinesq extension.

New interaction solutions of periodic waves and solitary waves for the (n+1)-dimensional double sinh-Gordon equation

New exact solutions with an arbitrary function for the (n+1)-dimensional double sinh-Gordon equation are studied by means of auxiliary solutions of the cubic nonlinear Klein–Gordon (CNKG) fields. By a proper selection of the arbitrary function and the appropriate solutions of the CNKG systems, new wave solutions including periodic–kink like waves, periodic–solitoffs and periodic waves are obtained explicitly.