Exact internal waves of a Boussinesq system

We consider a Boussinesq system describing one-dimensional internal waves which develop at the boundary between two immiscible fluids, and we restrict to its traveling waves. The method which yields explicitly all the elliptic or degenerate elliptic solutions of a given nonlinear, any order algebraic ordinary differential equation is briefly recalled. We then apply it to the fluid system and, restricting in this preliminary report to the generic situation, we obtain all the solutions in that class, including several new solutions.Comment: 11 pages, Waves and stability in continuous media, Palermo, 28 June-1 July 2009. Eds. A.Greco, S.Rionero and T.Ruggeri (World scientific, Singapore, 2010

Singularites in the Bousseneq equation and in the generalized KdV equation

In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical model of some nonlinear wave problems

An auxiliary ordinary differential equation and the exp-function method

AbstractIn this paper, the new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary ordinary differential equation are derived by using exp-function method, and then the exact solutions of the nonlinear evolution equations are derived with aid of the auxiliary ordinary differential equation. As examples, the classical KdV equation, Boussinesq equation, (3+1)-dimensional Jimbo–Miwa equation and Benjamin–Bona–Mahony equation are discussed and the exact solutions are derived

Application of the Extended G\u27/G-expansion Method to the Improved Eckhaus Equation

In this paper, the extended (G\u27/G)-expansion method is used to seek more general exact solutions of the improved Eckhaus equation and the (2+1)-dimensional improved Eckhaus equation. As a result, hyperbolic function solutions, trigonometric function solutions and rational function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions are also derived from the traveling wave solutions. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics

Elliptic solutions and solitary waves of a higher order KdV--BBM long wave equation

We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ functions of both third and fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV--BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstrass elliptic $\wp$ functions. For the fifth-order KdV--BBM equation we show that a parameter $\gamma=\frac {1}{12}$, for which the equation has a Hamiltonian, represents a restriction for which there are constraint curves that never intersect a region of unbounded solitary waves, which in turn shows that only dark or bright solitons and no unbounded solutions exist. Motivated by the lack of a Hamiltonian structure for $\gamma\neq\frac{1}{12}$ we develop $H^k$ bounds, and we show for the non Hamiltonian system that dark and bright solitons coexist together with unbounded periodic solutions. For nonzero boundary conditions, due to the complexity of the nonlinear algebraic system of coefficients of the elliptic equation we construct Weierstrass solutions for a particular set of parameters only.Comment: 13 pages, 6 figure

Solitary and Periodic Exact Solutions Of the Viscosity-capillarity van der Waals Gas Equations

Periodic and soliton solutions are derived for the (1+1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. The system is handled via the e-φ(ξ) -expansion method. The obtained solutions have been articulated by the hyperbolic, trigonometric, exponential and rational functions with arbitrary constants. Mathematical analysis and numerical graphs are provided for some solitons, periodic and kink solitary wave solutions to visualize the dynamics of equations. Obtained results reveal that the method is very influential and effective tool for solving nonlinear partial differential equations in applied mathematics