Juan-Ming Yuan

A Dual-Petrov-Galerkin Method for the Kawahara-Type Equations

Abstract An efficient and accurate numerical scheme is proposed, analyzed and implemented for the Kawahara and modified Kawahara equations which model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. The scheme consists of dual-Petrov-Galerkin method in space and Crank-Nicholson-leap-frog in time such that at each time step only a sparse banded linear system needs to be solved. Theoretical analysis and numerical results are presented to show that the proposed numerical is extremely accurate and efficient for Kawahara type equations and other fifth-order nonlinear equations.

The effect of dissipation on solutions of the complex KdV equation

It is known that some periodic solutions of the complex KdV equation with smooth initial data blow up in finite time. In this paper, we investigate the effect of dissipation on the regularity of solutions of the complex KdV equation. It is shown here that if the initial datum is comparable to the dissipation coefficient in the L2-norm, then the corresponding solution does not develop any finite-time singularity. The solution actually decays exponentially in time and becomes real analytic as time elapses. Numerical simulations are also performed to provide detailed information on the behavior of solutions in different parameter ranges.

The generalized Buckley-Leverett and the regularized Buckley-Leverett equations

This paper studies solutions of the generalized Buckley-Leverett (BL) equation with variable porosity and solutions of the regularized BL equation with the Burgers-Benjamin-Bona-Mahony-type regularization. We construct global in time weak solutions to the Cauchy problem and to the initial- and boundary-value problem for the generalized BL equation and global classical solutions for the regularized BL equation. Solutions of the regularized BL equation are shown to converge to the corresponding solution of the generalized BL equation when the coefficient γ of the BBM term and the coefficient ν of the Burgers term obey γ = O(ν2).

A new solution representation for the BBM equation in a quarter plane and the eventual periodicity

The initial-boundary-value problem for the Benjamin–Bona–Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behaviour (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. Towards this end, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator ∂t +α∂x − γ∂ xxt defined in the space–time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase. The eventual periodicity of the full BBM equation is obtained under a suitable assumption imposed on its solution.

Fifth-order complex Korteweg?de Vries-type equations

This paper studies spatially periodic complex-valued solutions of the fifth-order Korteweg?de Vries (KdV)-type equations. The aim is at several fundamental issues including the existence, uniqueness and finite-time blowup problems. Special attention is paid to the Kawahara equation, a fifth-order KdV-type equation. When a Burgers dissipation is attached to the Kawahara equation, we establish the existence and uniqueness of the Fourier series solution with the Fourier modes decaying algebraically in terms of the wave numbers. We also examine a special series solution to the Kawahara equation and prove the convergence and global regularity of such solutions associated with a single mode initial data. In addition, finite-time blowup results are discussed for the special series solution of the Kawahara equation.

Complex-Valued Burgers and KdV-Burgers Equations

Spatially periodic complex-valued solutions of the Burgers and KdV–Burgers equations are studied in this paper. It is shown that for any sufficiently large time T, there exists an explicit initial datum such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions.

The Kawahara equation in weighted Sobolev spaces

The initial- and boundary-value problem for the Kawahara equation, a fifth-order KdV type equation, is studied in weighted Sobolev spaces. This functional framework is based on the dual-Petrov–Galerkin algorithm, a numerical method proposed by Shen (2003 SIAM J. Numer. Anal. 41 1595–619) to solve third and higher odd-order partial differential equations. The theory presented here includes the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. If the L2-norm of the initial data is sufficiently small, these solutions decay exponentially in time. Numerical computations are performed to complement the theory.