Xiao-yan Tang

Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems

The multilinear variable separation approach and the related “universal” formula have been applied to many (2+1)-dimensional nonlinear systems. Starting from the universal formula, abundant (2+1)-dimensional localized excitations have been found. In this paper, the universal formula is extended in two different ways. One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the (2+1)-dimensional symmetric sine-Gordon system. The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved. Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons. The FSWs...

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.

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.

Equations of arbitrary order invariant under the Kadomtsev-Petviashvili symmetry group

By means of a simple new approach, a general Kadomtsev–Petviashvili (KP) family with an arbitrary function of group invariants of arbitrary order is proposed. It is proved that the general KP family possesses a common infinite dimensional Kac–Moody–Virasoro Lie point symmetry algebra. The known fourth order one can be re-obtained as a special example. The finite transformation group is presented in a clearer form. The Kac–Moody–Virasoro group invariant solutions and the Kac–Moody group invariant solutions of the KP family are determined by the Boussinesq and KdV families, respectively.

Infinitely many generalized symmetries and Painlevé analysis of a (2 + 1)-dimensional Burgers system

Infinitely many generalized symmetries of a coupled (2 + 1)-dimensional Burgers system are obtained by means of the formal series symmetry approach. It is found that the generalized symmetries constitute a closed infinite-dimensional Lie algebra. Three interesting special cases are presented, including a closed infinite-dimensional Lie algebra and a Kac–Moody–Virasoro-type Lie symmetry algebra. From the first one of the positive flow, a new integrable coupled system of the modified Korteweg–de Vries equation and the potential Boiti–Leon–Manna–Pempinelli equation is constructed. In addition, it is demonstrated that the coupled Burgers system can pass the Painleve test.

Child-Langmuir flow in a planar diode filled with charged dust impurities

The Child–Langmuir (CL) flow in a planar diode in the presence of stationary charged dust particles is studied. The limiting electron current density and other diode properties, such as the electrostatic potential, the electron flow speed, and the electron number density, are calculated analytically. A comparison of the results with the case without dust impurities reveals that the diode parameters mentioned above decrease with the increase of the dust charge density. Furthermore, it is found that the classical scaling of D−2 (the gap spacing D) for the CL current density remains exactly valid, while the scaling of V3∕2 (the applied gap voltage V) can be a good approximation for low applied gap voltage and for low dust charge density.

Modulational instability and exact solutions of the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation

The modulational instability (MI) of the coupled nonlinear Schrodinger and nonlinear Klein–Gordon equations is investigated. It is found that there are a number of possibilities for the MI regions due to the generalized dispersion relation, which relates the frequency and wavenumber of the modulating perturbations. Some exact travelling wave solutions are constructed via the solutions of a 4 model through a simple mapping relation. Furthermore, we present five different types of solutions representing possible final states of modulationally unstable perturbations. The profiles of solitary wave structures are displayed for some fixed parameters.

Nanopteron solution of the Korteweg-de Vries equation

The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical studies for many decades. However, the analytical solution of such a special soliton is rarely considered. In this letter, we study the explicit nanopteron solution of the Korteweg-de Vries (KdV) equation. Starting from the soliton-cnoidal wave solution of the KdV equation, the nanopteron structure is shown to exist. It is found that for the suitable choice of the wave parameters, the soliton core of the soliton-cnoidal wave trends to be a classical soliton of the KdV equation and the surrounded cnoidal periodic wave appears as small amplitude sinusoidal variations on both sides of the main core. Some interesting features of the wave propagation are revealed. In addition to the elastic interaction, it is surprising that the phase shift of the cnoidal periodic wave after the interaction with the soliton core is always half its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.

Symmetry analysis and similarity electrostatic waves in a nonuniform dusty magnetoplasma

We apply the group theory to a (3+1)-dimensional nonlinear system relevant for the low-frequency electrostatic waves in a nonuniform dusty magnetoplasma. In correspondence with the generators of the symmetry group allowed by the system, new types of similarity reductions are performed. Some new exact solutions are obtained, which can be in the form of solitary waves, shock waves and periodic waves. Especially, our solutions indicate that the system may have time-dependent nonlinear shears. Some explicit similarity electrostatic wave solutions with time periodic nonlinear shears are displayed graphically.