Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability

Forced oscillation of viscous Burgers' equation with a time-periodic force

“…Existence of periodic solutions of damped KdV equation on half line is proved with similar fashion in a variant of the Y t,T as well in [2]. Moreover, local and global stability of periodic solutions of KdV or KdV-Burgers equation posed on finite domains were obtained by one of us in [16,17,20]. The outline of the proof is, separating the problem into two parts: linear and nonlinear, and linear subproblem is done immediately while one is able to employ contractioon mappings to obtain boundedness of solution in Y t,T for the nonlinear subproblem with small bound, then asymptotic periodicity with exponential decay can be proved.…”

“…Readers may refer to R. Temam's book [15] for Galerkin's decomposition and corresponding wellposedness framework applied to dissipative equations. Besides, KdV (Korteweg-de Vries) equations are proved to possess periodic solutions by one of the authors, if similar assumption that the force has small amplitude holds (see, e.g., [16,17,20]). Instead of Galerkin's method as solution of KdV equation doesn't have form of Galerkin-type decomposition, the proof is mainly based on one of local wellposedness results in a particular Sobolev space Y t,T (defined on time interval [t, t + θ] with a particular T including smoothing effects; see next section) in J.…”

Periodic and almost periodic solutions for the damped Korteweg‐de Vries equation

“…The above results considered the KdV equation in unbounded domain or , however, in applications, we may investigate the propagation of water waves in a bounded channel. So it is worthy to consider the KdV equation on a bounded domain; Georgiev investigated the periodic solution of the KdV equation in a bounded domain without any boundary condition, Usman and Zhang obtained the existence of the periodic solutions of the KdV equation with boundary force.…”

Nonlinearly Self-Adjoint, Conservation Laws and Solutions for a Forced BBM Equation

“…Equation (1) was introduced to model long waves in nonlinear dispersive systems. Some special cases of (1) are studied in [2,3]. If = 1/2 and = 1, = 0, = 0, and = 0, then (1) reduces to the celebrated Benjamin-Bona-Mahony (BBM) equation…”

“…The aim of this paper is to prove that (3) is nonlinearly self-adjoint. We determine, by using the Lie generators of (3) and the notation and techniques of [12], some nontrivial conservation laws for (3). Finally, we present some exact solutions for a special case of (3).…”