Energy decay for the modified Kawahara equation posed in a bounded domain

Studied here is the eventual dissipation of solutions to initial–boundary value problems for the modified Kawahara equation with and without a localized damping term included. It is shown that solutions of undamped problems posed on a bounded interval may not decay if the length of the interval is critical. In contrast, the energy associated to the locally damped problems is shown to be exponentially decreased independently of the interval length.     

Global regular solutions for the 3D Kawahara equation posed on unbounded domains

An initial boundary value problem for the 3D Kawahara equation posed on a channel-type domain was considered. The existence and uniqueness results for global regular solutions as well as exponential decay of small solutions in the H²-norm were established.     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain II

Studied here is an initial- and boundary-value problem for the Korteweg-de Vries equation

     

A non-homogeneous boundary-value problem for the KdV–Burgers equation posed on a finite domain

We consider the initial–boundary value problem for the KdV–Burgers equation posed on a bounded interval $[a,b]$. This problem features non-homogeneous boundary conditions applied at $x=a$ and $x=b$ and is known to have unique global smooth solution.     

Ill-posedness of Kawahara equation and Kaup-Kupershmidt equation

In this paper we consider the Cauchy problems for the Kawahara equation and the Kaup-Kupershmidt equation. By using the general well-posedness principle introduced by I. Bejenaru and T. Tao (2006) [1], we prove that the Kawahara equation is ill-posed for the initial data in Hs(R) with and the Kaup-Kupershmidt equation is ill-posed for the initial data in Hs(R) with .     

Doubly periodic solutions of the modified Kawahara equation

Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors.     

Well-posedness of Korteweg-de Vries-Burgers equation on a finite domain

In this paper,we consider the Korteweg-de Vries-Burgers equation on a finite domain with initial value and nonhomogeneous boundary conditions. This particular problem arises in the theory of ferroelectricity. We first get the local well-posedness of the problem, and then under the help of the local result, we use nonlinear interpolation to have the global well-posedness of the problem.     

Solitary wave solution for the generalized Kawahara equation

The travelling wave ansatz is used to find the solitary wave solution of the generalized Kawahara equation. The ansatz is obtained from the structure of the soliton solution of the Kawahara equation and the modified Kawahara equation. The first two integrals of motion of the generalized Kawahara equation are also computed in this work.     

On the Cauchy problem for the generalized Kawahara equation

We consider global weak solutions of the Cauchy problem for the generalized Kawahara equation. We prove theorems on the existence, uniqueness, and interior regularity as well as the exponential decay for large time.     

Mixed problem for the Kawahara equation in a half-strip

We prove the global well-posedness of the mixed problem for the Kawahara equation in a half-strip under natural conditions on the boundary data.     

Existence of a family of soliton-like solutions for the Kawahara equation

Existence is proved for a family of soliton-like solutions for the nonlinear evolution equation u_t–+uu_x+u_xxx-u_xxxxx=O. The problem is reduced to investigating the fixed points of the operator

On the radius of spatial analyticity for the modified Kawahara equation on the line

First, by using linear and trilinear estimates in Bourgain type analytic and Gevrey spaces, the local well‐posedness of the Cauchy problem for the modified Kawahara equation on the line is established for analytic initial data that can be extended as holomorphic functions in a strip around the x‐axis. Next we use this local result and a Gevrey approximate conservation law to prove that global solutions exist. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity given by , where can be taken arbitrarily small and c is a positive constant.     

Strichartz estimates for dispersive equations and solvability of the Kawahara equation

We first establish a series of Strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by Kenig, Ponce and Vega. Next we use such estimates to study solvability of the Cauchy problem of the Kawahara equation in the class C(R,Hs(R)). Local existence is proved for s>1/4 and global existence is proved for s?2.     

A note on new exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, New Exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.     

New exact solutions for the Kawahara equation using Exp-function method

In this paper, we applied the Exp-function method to solve the Kawahara equation. This method can be used to obtain new exact solutions and periodic solutions with parameters are obtained. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful (mathematical tools) for discrete nonlinear evolution equations in mathematical physics.     

Boundary equations in the finite transfer method for solving differential equation systems

The finite transfer method is going to be used to solve a p system of linear ordinary differential equations. The complete problem is extended by adding the p boundary equations involved. It is chosen a fourth order scheme to obtain finite transfer expressions. A recurrence strategy is used in these equations and permits one to relate different points in the domain where boundary equations are defined. Finally a 2p algebraic system of equations is noted and solved. To show the efficiency and accuracy, the method is applied to determine the structural behavior of a bending beam with different supports and to solve a differential equation of second degree with different boundary conditions.