Extended F-Expansion Method and Periodic Wave Solutionsfor Klein-Gordon-Schrödinger Equations

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrödinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

Bäcklund Transformation and Multisoliton Solutions in Terms of Wronskian Determinant for (2+1)-Dimensional Breaking Soliton Equations with Symbolic Computation

In this paper, two types of the (2+1)-dimensional breaking soliton equations are investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilinear forms and Bäcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilinear equations. Via the Wronskian technique, it is proved that theBäcklund transformations obtained are the ones between the (N-1)- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonic properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.     

Single Peak Soliton and Periodic Cusp Wave of the Generalized Schrödinger-Boussinesq Equations

In this paper, we study peakon, cuspon, smooth soliton and periodic cusp wave of the generalized Schrödinger-Boussinesq equations. Based on the method of dynamical systems, the generalized Schrödinger-Boussinesq equations are shown to have new the parametric representations of peakon, cuspon, smooth soliton and periodic cusp wave solutions. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

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 Schrödinger (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.     

Carmeli's Gravitational Field Equations

Carmeli has proposed spinorial field equations in curved space-time to describe gravitation. In this paper we give the relationship between these equations and the standard Einstein gravitational field equations. In particular we show that all solutions to Einstein's equations are solutions to Carmeli's equations, but not vice versa.     

非线性耦合微分方程组的精确解析解

提出了利用耦合的Riccati方程组的某些特解构造非线性微分方程组精确解析解的一种方法.应用这种方法研究了两个耦合的常微分方程组,系统地获得了它们的一些精确解.给出了非线性浅水波近似方程组和非线性Schr?dinger-KdV方程组若干新的孤波解. 关键词： 非线性耦合方程组 Riccati方程组 符号计算 孤波     

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space—time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.     

On the investigation of nonlinear waves of the Hirota and the Maxwell–Bloch equation in nonlinear optics

In this paper, considering the Hirota and Maxwell-Bloch (H-MB) equations which is governed by femtosecond pulse propagation through two-level doped fibre system, we construct the Darboux transformation of this system through linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.     

Yang-Mills fields with nonquadratic energy

A gauge-invariant nonlinear Hodge-de Rham system is introduced. These equations have the same relation to the Yang-Mills equations that the conventional nonlinear Hodge equations have to the equations of classical Hodge theory. Conditions are given under which weak solutions are locally Hölder continuous. The existence of solutions is proven for variational points of a certain class of nonquadratic energy functionals.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

Bäcklund Transformat ion and Mult isoliton-Like Solutions for (2+1)-Dimensional Dispersive Long Wave Equations

A Bäcklund transformation of the (2+1)-dimensional dispersive long wave equations is derived by using the developed homogeneous balance method. by means of the Bäcklund transformation, the new multisoliton-like solution and other two types of exact solutions to these equations are constructed.     

Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

In this paper, the formulation of time-fractional (TF) electrodynamics is derived based on the Riemann-Silberstein (RS) vector. With the use of this vector and fractional-order derivatives, one can write TF Maxwell’s equations in a compact form, which allows for modelling of energy dissipation and dynamics of electromagnetic systems with memory. Therefore, we formulate TF Maxwell’s equations using the RS vector and analyse their properties from the point of view of classical electrodynamics, i.e., energy and momentum conservation, reciprocity, causality. Afterwards, we derive classical solutions for wave-propagation problems, assuming helical, spherical, and cylindrical symmetries of solutions. The results are supported by numerical simulations and their analysis. Discussion of relations between the TF Schrödinger equation and TF electrodynamics is included as well.     

(2+1)-dimensional coupled Boussinesq equations for Rossby waves in two-layer cylindrical fluid*

In this paper, the existence and propagation characteristics of Rossby waves in a two-layer cylindrical fluid are studied. Firstly, based on the dimensionless baroclinic quasi-geostrophic vortex equations including exogenous and dissipative, we derive new (2+1)-dimensional coupled Boussinesq equations describing wave propagation in polar coordinates by employing a multiscale analysis and perturbation method. Then, the Lie symmetries and conservation laws of the coupled Boussinesq equations are analyzed. Subsequently, by using the $(G^{\prime} /G)$-expansion method, the exact solutions of the (2+1)-dimensional coupled Boussinesq equations are obtained. Finally, the effects of coupling term coefficients on the propagation characteristics of Rossby waves are analyzed.     

The Cartesian analytical solutions for the N-dimensional compressible Navier–Stokes equations with density-dependent viscosity

In this paper, we prove the existence of general Cartesian vector solutions u = b(t) + A(t)x for the N-dimensional compressible Navier–Stokes equations with density-dependent viscosity, based on the matrix and curve integration theory. Two exact solutions are obtained by solving the reduced systems.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions are performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

用三角函数法获得非线性Boussinesq方程的广义孤子解

找到一个合适的代换——三角函数法，将非线性Boussinesq微分方程转换为非线性代数方程组.用吴消元法求解该非线性代数方程组，从而获得一般形式Boussinesq微分方程的广义孤子解. 关键词： Boussinesq方程 吴消元法 非线性代数方程组 孤子解     

Non‐Volkov solutions for a charge in a plane wave

As it is known, a set of solutions of the Klein‐Gordon and Dirac equations with a plane‐wave field was found for the first time by Volkov. We construct new solutions of these equations different from the Volkov ones. In particular, the new solutions are characterized by quantum numbers different from Volkov solutions. In fact, our result is based on the demonstration that the transversal charge motion in a plane wave can be mapped by a special quantum transformation to transversal free particle motion. Similarly, we find new sets of solutions of the Klein‐Gordon and Dirac equations with the combined electromagnetic field.     

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.