Higher Order Nonlinear Evolution Equations Research Articles

Small-stencil Padé schemes to solve nonlinear evolution equations

A set of small-stencil new Pade schemes with the same denominator are presented to solve high-order nonlinear evolution equations. Using this scheme, the fourth-order precision can not only be kept, but also the final three-diagonal discrete systems are solved by simple Doolittle methods, or ODE systems by Runge-Kutta technique. Numerical samples show that the schemes are very satisfactory. And the advantage of the schemes is very clear compared to other finite difference schemes.

Higher order nonlinear evolution equations for a three-dimensional electromagnetic wave packet in a plasma

Starting from the full set of governing equations including the weak relativistic effect, three coupled nonlinear equations with higher older nonlinear and dispersive terms are derived, which describe the nonlinear evolution of a three-dimensional electromagnetic wave packet propagating in a plasma. Two particular cases are considered, where the space dependence of the dependent variables is either on the space coordinate along the direction of propagation of the wave or on that along an arbitrary fixed direction. In both cases the three coupled equations reduce to a single equation. This single equation is used to study the modulational instability of a uniform electromagnetic wave train. It is found that due to the inclusion of the weak relativistic effect there is always modulational instability, and a higher order nonlinear term in the evolution equation has a stabilizing influence.

Groups of scalings and invariant sets for higher-order nonlinear evolution equations

Consider a nonlinear $m^{\text{th}}$ order evolution PDE \begin{equation} u_t = \mathbf A(u) \equiv A(x,u,u_1, \dots ,u_m), \quad u=u(x,t), \quad (x,t) \in Q=(0,1) \times [0,1], \tag*{(*)} \end{equation} where $A$ is a $C^\infty$ function and $u_t = \partial u/\partial t$, $u_i = \partial ^i u / \partial x^i$. If (*) is invariant under a group of scalings with the infinitesimal generator $ X =x \frac {\partial}{\partial x} + \mu t \frac {\partial}{\partial t} $ ($\mu$ is a constant ``scaling order" of $\mathbf A$), then the PDE admits exact self-similar solutions depending on the single invariant variable $u(x,t) = {\theta } (\xi)$, $ \xi = x/t^{1/\mu}$, where ${\theta }$ solves a nonlinear $m^{\text{th}}$ order ODE associated with the PDE. We prove that when the operator $\mathbf A$ is composed of a finite sum of operators with different scaling orders, $\mathbf A = \sum \mathbf A_i$, and no group of scalings exists, the exact solutions can be constructed via the invariance of the set $S_0 = \{u: u_1 =F(u)/x\}$ of a contact first-order differential structure, where $F$ is a smooth function to be determined. The time-evolution on $S_0$ is shown to be governed by a first-order dynamical system. We thus observe that besides scaling group properties, the invariance of $S_0$ specifies new sets of solutions described by first-order ODEs. The approach applies to a class of nonlinear parabolic equations of the second and of the fourth order.

On the derivation of first integrals for similarity solutions

In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equation. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.