Sen-Yue Lou

Revisitation of the localized excitations of the (2+1)-dimensional KdV equation

In a previous paper (Lou S-y 1995 J. Phys. A: Math. Gen. 28 7227), a generalized dromion structure was revealed for the (2+1)-dimensional KdV equation, which was first derived by Boiti et al (Boiti M, Leon J J P, Manna M and Pempinelli F 1986 Inverse Problems 2 271) using the idea of the weak Lax pair. In this paper, using a Backlund transformation and the variable separation approach, we find there exist much more abundant localized structures for the (2+1)-dimensional KdV equation. The abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution. Some special types of dromion solution, lumps, breathers, instantons and the ring type of soliton, are discussed by selecting the arbitrary functions appropriately. The dromion solutions can be driven by sets of straight-line and curved-line ghost solitons. The dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves. The breathers may breathe both in amplitude and in shape.

Nonlocal symmetries related to Bäcklund transformation and their applications

Starting from nonlocal symmetries related to Backlund transformation (BT), many interesting results can be obtained. Taking the well-known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in an elegant and compact form which comes from BT is presented and used to perform research works in two main subjects: the nonlocal symmetry is localized by introducing suitable and simple auxiliary-dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions; some other models both in finite and infinite dimensions are generated under new nonlocal symmetry. The finite-dimensional models are completely integrable in Liouville sense, which are shown equivalent to the results given through the nonlinearization method for Lax pair.

Coupled KdV equations derived from two-layer fluids

Some types of coupled Korteweg de-Vries (KdV) equations are derived from a two-layer fluid system. In the derivation procedure, an unreasonable y-average trick (usually adopted in the literature) is removed. The derived models are classified by means of the Painleve test. Three types of τ-function and multiple soliton solutions of the models are explicitly given via the exact solutions of the usual KdV equation. It is also discovered that a non-Painleve integrable coupled KdV system can have multiple soliton solutions.

Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method

A new direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Using the direct method for the well-known (2+1)-dimensional Kadomtsev–Petviashvili equation and the Ablowitz–Kaup–Newell–Segur system, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.

Variable separation approach for a differential-difference system: special Toda equation

A bilinear variable separation approach is used to construct some special solutions for a differential-difference Toda equation. The semi-discrete form of the continuous formula which describes some types of special solutions for many (2 + 1)-dimensional continuous systems is found for a suitable quantity of the differential-difference Toda equation. Thus abundant semi-discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions.

Abundant coherent structures of the dispersive long-wave equation in (2+1)-dimensional spaces

Abstract By means of the variable separation approach, the abundant localized coherent structures of a (2+1)-dimensional dispersive long-wave equation (2DDLWE) are given out. The result formula for the coherent solutions is totally the same as that of the asymmetric Nizhnik–Novikov–Veselov equation and the asymmetric Davey–Stewartson equation. Especially, from the figure plots of three-soliton solution, we find that the interaction among the travelling saddle type ring solitons is elastic.

(2 + 1)-dimensional compacton solutions with and without completely elastic interaction properties

Taking the (2 + 1)-dimensional Broer–Kaup–Kupershmidt system as a simple example, some special types of (2 + 1)-dimensional compacton solutions are constructed. It is shown that there is quite rich interaction behaviour between two travelling compactons. For some types of compactons, the interactions among them may not be completely elastic. For some others, the interactions are completely elastic. There is no phase shift for the interactions of the (2 + 1)-dimensional compactons discussed in this paper.

Localized solutions with chaotic and fractal behaviours in a (2+1)-dimensional dispersive long-wave system

A substrate serving as a heat sink for a semiconductor efficiently radiates heat from a semiconductor element mounted thereon. The substrate consists of a composite alloy metal which consists of a sintered body of a metal powder having a high melting point such as W and Mo impregnated with a filling metal such as Cu and Ag, wherein the sintered body of a metal powder having a high melting point has a grain size composition of a combination of a plurality of powder groups having statistically different average grain sizes from group to group, and the powder of each group is dispersed uniformly.

(2+1)-dimensional (M+N)-component AKNS system: Painlevé integrability, infinitely many symmetries, similarity reductions and exact solutions

The (2+1)-dimensional (M+N)-component AKNS system that is derived from the inner parameter dependent symmetry constraint of the KP equation is studied in detail. First, the Painleve integrability of the model is proved by using the standard WTC and Kruskal approach. Using the formal series symmetry approach, the generalized KMV symmetry algebra and the related symmetry group are found. The two-dimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized KMV symmetry algebra and the direct method. Abundant localized coherent structures are revealed by the variable separation approach. Some special types of the localized excitations like the multiple solitoffs, dromions, lumps, ring solitons, breathers and instantons are plotted also.

New variable separation approach: application to nonlinear diffusion equations

The concept of the derivative-dependent functional separable solution (DDFSS), as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the DDFSS is obtained and some exact solutions to the resulting equations are described.