Exact solutions and localized excitations of a （3＋ 1）-dimensional Gross-Pitaevskii system

Periodic wave solutions and solitary wave solutions to a generalized (3+1)-dimensional Gross-Pitaevskii equation with time-modulated dispersion, nonlinearity, and potential are derived in terms of an improved homogeneous balance principle and a mapping approach. These exact solutions exist under certain conditions via imposing suitable constraints on the functions describing dispersion, nonlinearity, and potential. The dynamics of the derived solutions can be manipulated by prescribing specific time-modulated dispersions, nonlinearities, and potentials. The results show that the periodic waves and solitary waves with novel behaviors are similar to the similaritons reported in other nonlinear systems.     

Wronskian and Grammian solutions for the （3＋ 1）-dimensional Jimbo-Miwa equation

<正>In this paper,based on Hirota’s bilinear method,the Wronskian and Grammian techniques,as well as several properties of the determinant,a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented.They guarantee that the Wronskian determinant and the Grammian determinant solve the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form.Then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions.At last,with the aid of Maple,some of these special exact solutions are shown graphically.     

Lie group analysis,numerical and non-traveling wave solutions for the(2+1)-dimensional diffusion–advection equation with variable coefficients

In this paper, the variable-coefficient diffusion–advection(DA) equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended(G /G)-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.     

Exact solutions of nonlinear fractional differential equations by （G＇/G）-expansion method

In this paper,we use the fractional complex transform and the(G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions.The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie’s modified Riemann–Liouville derivative into its ordinary differential equation.It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.     

Novel interaction phenomena of the(3+1)-dimensional Jimbo–Miwa equation

In this paper,we give the general interaction solution to the(3+1)-dimensional Jimbo–Miwa equation.The general interaction solution contains the classical interaction solution.As an example,by using the generalized bilinear method and symbolic computation by using Maple software,novel interaction solutions under certain constraints of the(3+1)-dimensional Jimbo–Miwa equation are obtained.Via three-dimensional plots,contour plots and density plots with the help of Maple,the physical characteristics and structures of these waves are described very well.These solutions greatly enrich the exact solutions to the(3+1)-dimensional Jimbo–Miwa equation found in the existing literature.     

New localized excitations in a (2+1)-dimensional Broer—Kaup system

Starting with the extended homogeneous balance method and a variable separation approach, a general variable separation solution of the Broer—Kaup system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakon and fractal localized solutions, some new types of localized excitations, such as compacton and folded excitations, are obtained by introducing appropriate lower-dimensional piecewise smooth functions and multiple-valued functions, and some interesting novel features of these structures are revealed.     

New exact periodic solutions to （2＋1）-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.     

Spatiotemporal self-similar solutions for the nonautonomous （3＋1）-dimensional cubic-quintic Gross-Pitaevskii equation

With the help of similarity transformation,we obtain analytical spatiotemporal self-similar solutions of the nonautonomous(3+1)-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction,nonlinearity,harmonic potential and gain or loss when two constraints are satisfied.These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction,nonlinearity and the gain/loss.Based on these analytical results,we investigate the dynamic behaviours in a periodic distributed amplification system.     

Comparative study of travelling-wave and numerical solutions for the coupled short pulse （CSP） equation

The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find reductions in terms of system of ordinary differential equations, and exact solutions of the CSP equation are derived, which are compared with numerical solutions using the classical fourth-order Runge-Kutta scheme.     

Explicit and exact travelling plane wave solutions of the （2＋1）—dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions,periodic wave solutions,Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein－Gordon equation.     

Exotic interactions between solitons of the （2＋1）-dimensional asymmetric Nizhnik-Novikov-Veselov system

Starting from the extended tanh-function method (ETM) based on the mapping method, the variable separation solutions of the (2+1)-dimensional asymmetric Nizhnik--Novikov--Veselov (ANNV) system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semi-foldon, are investigated.     

Painlevé property,local and nonlocal symmetries,and symmetry reductions for a (2+1)-dimensional integrable KdV equation

The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the B?cklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite B?cklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.     

The transverse laser induced thermoelectric voltages in step flow growth （1-x）Pb(Mg1/3Nb2/3)O3-xPbTiO3 thin films

This paper reports that the transverse laser induced thermoelectric voltages (LITV) are observed for the first time in the step flow growth (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-PT,x = 0.20, 0.33, 0.50) thin films deposited on vicinal-cut strontium titanate single crystal substrates. Because lead magnesium niobate-lead titanate is a solid solution of lead magnesium niobate (PMN) and lead titanate (PT), there are two types of signals. One is wide with a time response of a microsecond, and the other superimposed with the wide signal is narrow with a time response of a nanosecond. The transverse LITV signals depend on the ratio of PMN to PT drastically. Under the irradiation of 28-ns pulsed KrF excimer laser with the 248-nm wavelength,the largest induced voltage is observed in the 0.50Pb(Mg1/Nb2/3)O3-0.50PbTiO3 films. Moreover, the effects of film thickness, substrates, and tilt angles of substrates are also investigated.