Exact Solution of Space-Time Fractional Coupled EW and Coupled MEW Equations Using Modified Kudryashov Method

In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation(CEWE) and the space-time fractional coupled modified equal width wave equation(CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.     

Variable Separation and Exact Solutions to Generalized Nonlinear Diffusion Equations

We study in detail a method to find the generalized nonlinear diffusion equations,which can be solved by means of the variable separation approach.A complete list of canonical forms for such equations,which admit the functional separable solutions,is botained and some exact solutions to the resulting equations are described. A number of methods have been proven to be effective for finding symmetry reductions and constructing exact solutions to nonlinear diffusion equations.     

Exact solutions and residual symmetries of the Ablowitz–Kaup–Newell–Segur system

The residual symmetries of the Ablowitz–Kaup–Newell–Segur(AKNS)equations are obtained by the truncated Painleve′analysis.The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system.The local Lie point symmetries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries,which suggests that the residual symmetry method is a useful complement to the classical Lie group theory.The calculation on the symmetries shows that the enlarged equations are invariant under the scaling transformations,the space–time translations,and the shift translations.Three types of similarity solutions and the reduction equations are demonstrated.Furthermore,several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Bcklund transformations between the AKNS equations and the Schwarzian AKNS equation.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

Abstract In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f （ξ）, is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

Variable coefficient nonlinear systems derived from an atmospheric dynamical system

Variable coefficient nonlinear systems, the Korteweg de Vries (KdV), the modified KdV (mKdV) and the nonlinear Schr?dinger (NLS) type equations, are derived from the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane by means of the multi-scale expansion method in two different ways, with and without the so-called y-average trick. The non-auto-B\"acklund transformations are found to transform the derived variable coefficient equations to the corresponding standard KdV, mKdV and NLS equations. Thus, many possible exact solutions can be obtained by taking advantage of the known solutions of these standard equations. Further, many approximate solutions of the original model are ready to be yielded which might be applied to explain some real atmospheric phenomena, such as atmospheric blocking episodes.     

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Dark Soliton Solutions of Space-Time Fractional Sharma–Tasso–Olver and Potential Kadomtsev–Petviashvili Equations

Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the tanh ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.     

A KdV-Type Wronskian Formulation to Generalized KP,BKP and Jimbo–Miwa Equations

The purpose of this paper is to introduce a class of generalized nonlinear evolution equations, which can be widely applied to describing a variety of phenomena in nonlinear physical science. A Kd V-type Wronskian formulation is constructed by employing the Wronskian conditions of the Kd V equation. Applications are made for the(3+1)-dimensional generalized KP, BKP and Jimbo–Miwa equations, thereby presenting their Wronskian sufficient conditions.An N-soliton solution in terms of Wronskian determinant is obtained. Under a dimensional reduction, our results yield Wronskian solutions of the Kd V equation.     

Explicit solutions from residual symmetry of the Boussinesq equation

The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained: a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.     

Improved Speed and Shape of Ion-Acoustic Waves in a Warm Plasma

The basic set of fluid equations can be reduced to the nonlinear Kortewege-de Vries(KdV)and nonlinear Schro¨dinger(NLS)equations.The rational solutions for the two equations has been obtained.The exact amplitude of the nonlinear ion-acoustic solitary wave can be obtained directly without resorting to any successive approximation techniques by a direct analysis of the given field equations.The Sagdeev’s potential is obtained in terms of ion acoustic velocity by simply solving an algebraic equation.The soliton and double layer solutions are obtained as a small amplitude approximation.A comparison between the exact soliton solution and that obtained from the reductive perturbation theory are also discussed.     

Analytic Study of a Bose-Einstein Condensate in Waveguide with an Obstacle Potential

We investigate the exact solutions of one-dimensional （1D） time-independent Gross-Pitaevskii equation （GPE）, which governs a Bose-Einstein condensate （BEC） in the magnetic waveguide with a square-Sech potential. Both the bound state and transmission state are found and the corresponding spatial configurations and transport properties of BEC are analyzed. It is shown that the well-known absolute transmission of the linear system can occur in the considered nonlinear system.     

Representations and Classification of Traveling Wave Solutions to sinh-GSrdon Equation

Two concepts named atom solution and combinatory solution are defined. The classification of all single traveling wave atom solutions to sinh-Gordon equation is obtained, and qualitative properties of solutions are discussed. In particular, we point out that some qualitative properties derived intuitively from dynamic system method are not true. Finally, we prove that our solutions to sinh-Gordon equation include all solutions obtained in the paper [Z.T. Fu, et al., Commun. Theor. Phys. （Beijing, China） 45 （2006） 55]. Through an example, we show how to give some new identities on Jacobian elliptic functions.     

Analytical Solitary Wave Solutions to a （3＋1）-Dimensional Gross-Pitaevskii Equation with Variable Coefficients

A （3＋1）-dimensional Gross-Pitaevskii （GP） equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrodinger （NLS） equation. Exact solutions of the （3＋1）D GP equation are constructed via those of the NLS equation. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. Solitary and periodic wave solutions with snaking and breathing behavior are reported.     

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.     

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.     

Classification and Reduction of Generalized Thin Film Equations

Classification and reduction of the generalized fourth-order nonlinear differential equations arising from the liquid films are considered. It is shown that these equations have solutions on subspaces of the polynomial, exponential or trigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n = 4,..., 9. Several examples of exact solutions are presented.     

Fusion,fission, and annihilation of complex waves for the （2＋l）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system

With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Objective Reduction Solutions to Higher-Order Boussinesq System in （2＋1）-Dimensions

With the help of an objective reduction approach （ORA）, abundant exact solutions of （2＋1）-dimensional higher-order Boussinesq system （including some hyperboloid function solutions, trigonometric function solutions, and a rational function solution） are obtained. It is shown that some novel soliton structures, like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure, and kink-like multisoliton structure with actual physical meaning exist in the （2＋1）-dimensional higher-order Boussinesq system.     

Massive String Cosmology in Bianchi Type ⅢSpace-Time with Electromagnetic Field

Exact solution of Einstein＇s field equations is obtained for massive string cosmological model of Bianchi III space-time using the technique given by Letelier （1983） in presence of perfect fluid and electromagnetic field. To get the deterministic solution of the field equations the expansion 0 in the model is considered as proportional to the eigen value σ2^2of the shear tensor σi^j and also the fluid obeys the barotropic equation of state. It is observed that in early stage of the evolution of the universe string dominates over the particle whereas the universe is dominated by massive string at the late time. It is also observed that the string phase of the universe disappears in our model because particle density becomes negative. Some physical and geometric properties of the model are also discussed.     

Exact Solutions Expressible in Rational Formal Hyperbolic and Elliptic Functions for Nonlinear Differential-Difference Equation

A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.