Dissipative Behavior of Some Fully Non-Linear KdV-Type Equations

TL;DR

This paper investigates the dissipative behavior of certain non-linear KdV-type equations, showing through numerical evidence that solutions tend to a unique entropy solution as the parameter approaches zero, contrasting with classical dispersive cases.

Contribution

It demonstrates that non-linear g functions induce dissipative behavior in KdV-type equations, leading to convergence to entropy solutions, unlike the dispersive behavior in classical KdV.

Findings

Solutions exhibit dissipative behavior for non-linear g functions.

As the parameter δ approaches zero, solutions converge to entropy solutions.

Contrasts with classical dispersive KdV behavior.

Abstract

The KdV equation can be considered as a special case of the general equation u_{t} + f(u)_{x} - \delta g(u_{xx})_x = 0, \qquad \delta > 0, where f is non-linear and g is linear, namely $f(u)=u^2/2$ and g(v)=v. As the parameter $\delta$ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated . We show through numerical evidence that a completely different, dissipative behavior occurs when g is non-linear, namely when g is an even concave function such as $g(v)=-|v|$ or $g(v)=-v^2$. In particular, our numerical results hint that as $\delta -> 0$ the solutions converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.