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.     

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.     

Interaction between Soliton and Periodic Wave

A truncation for the Laurent series in the Padnleve analysis of the KdV equation is restudied. When the truncation occurs the singular manifold satisfies two compatible fourth-order PDEs, which are homogeneous of degree 3. Both of the PDEs can be factored in the operator sense. The common factor is a third-order PDE, which is homogeneous of degree 2. The first few Invariant manifolds of the third-order PDE are studied. We find that the invariant manifolds of the third-order PDE can be obtained by factoring the invariant manifolds of the KdV equation. A numerical solution of the third-order PDE facts about the third-order PDE. is also presented. The solution reveals some interesting     

Soliton and Periodic Travelling Wave Solutions for Quadratic Nonlinear Media

Many sets of the soliton and periodic travelling wave solutions for the quadratic χ^（2） nonlinear system are obtained by the Backlund transformation and the trial method. The property of the propagation for some travelling waves is investigated.     

Analytical Periodic Wave and Soliton Solutions of the Generalized Nonautonomous Nonlinear Schrödinger Equation in Harmonic and Optical Lattice Potentials

We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schrödinger equation with time- and space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain (loss) coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.     

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

Dynamical system theory is applied to the integrable nonlinear wave equation $u_t±(u^3−u^2)x+(u^3)xxx=0$. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation corresponds to the case of wave speed $c$=0. In the case of $c^6$≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.     

New Weierstrass Semi-rational Expansion Method to Doubly Periodic Solutions of Soliton Equations

Based on the Weierstrass elliptic function equation, a new Weierstrass semi-rational expansion method and its algorithm are presented. The main idea of the method changes the problem solving soliton equations into another one solving the corresponding set of nonlinear algebraic equations. With the aid of Maple, we choose the modified KdV equation, (2+1)-dimensional KP equation, and (3+1)-dimensional Jimbo-Miwa equation to illustrate our algorithm. As a consequence, many types of new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover the corresponding new Jacobi elliptic function solutions and solitary wave solutions are also presented as simple limits of doubly periodic solutions.     

Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities

We present a variational principle for small amplitude periodic solutions, with fixed frequency, of a completely resonant nonlinear wave equation. Existence and multiplicity results follow by min-max variational arguments. Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.     

The Periodic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobi elliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.     

On the Self-Similar Solutions of Generalized Hydrodynamics Equations and Nonlinear Wave Patterns

Abstract

Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated regimes, arising as a result with terms desribing relaxing and dissipative properties of the medium are described.     

Singular Wave Solutions of Two Integrable Generalized KdV Equations

We present some singular wave solutions such as multi-peaked periodic waves, multi-peaked kink waves, multi-peaked peakons as well as kink-compactons, associated with singular curves of generalized KdV equation and modified KdV equation. When a trajectory intersects with the singular curve, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed.     

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota-Satsuma KdV System

In this paper, we study the generalized coupled Hirota Satsuma KdV system by using the new generalizedtransformation in homogeneous balance method. As a result, many explicit exact solutions, which contain new solitarywave solutions, periodic wave solutions, and the combined formal solitary wave solutions, and periodic wave solutions,are obtained.     

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.     

A Generalized Representation Formula for Systems of Tensor Wave Equations

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski (Hyperbolic Equ. 4(3):401–433, 2007) to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization of (Hyperbolic Equ. 4(3):401–433, 2007) is a key component for extending Klainerman and Rodnianski’s breakdown criterion result for Einstein-vacuum spacetimes in (J. Amer. Math. Soc. 23(2):345–382, 2009) to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.     

Periodic Wave Solutions of Generalized Derivative Nonlinear Schrödinger Equation

A Darboux transformation of the generalized derivative nonlinear Schrodinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schrodinger equation are explicitly given.     

Doubly Periodic Propagating Wave for （2d-1）-Dimensional Breaking Soliton Equation

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the （2＋1）-dimensional breaking soliton system. By introducing Jacobi elliptic functions in the seed solution, two families of doubly periodic propagating wave patterns are derived. We investigate these periodic wave solutions with different modulus m selections, many important and interesting properties are revealed. The interaction of Jabcobi elliptic function waves are graphically considered and found to be nonelastic.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.