Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2001-02-17
Nonlinear Sciences
Exactly Solvable and Integrable Systems
32 pages, 1 figure, latex2e
Scientific paper
We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar function $\g$ that satisfies a scalar RH problem and a set of algebraic equations constrained by algebraic inequalities. The scalar function $\g$ is equivalent to the solution of the Lax-Levermore maximization problem. The solution of the set of algebraic equations satisfies the Whitham equations. We show that the scalar function $\g$ and the Lax-Levermore maximizer can be expressed as the solution of a linear overdetermined system of equations of Euler-Poisson-Darboux type. We also show that the set of algebraic equations and algebraic inequalities can be expressed in terms of the solution of a different set of linear overdetermined systems of equations of Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic equations is equivalent to the classical solution of the Whitham equations expressed by the hodograph transformation.
No associations
LandOfFree
Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-569136