Wave Number Shocks for the Tail of Korteweg-deVries Solitary Waves in Slowly Varying Media.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH
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Asymptotic solutions for the nonlinear, nonhomogeneous, Korteweg-deVries KdV partial differential equation with slowly varying coefficients are not in general uniformly valid. A uniform asymptotic expansion is obtained by finding separate expansions for different regions and matching. A KdV solitary wave propagating in slowly varying media is examined. Quasi-stationarity for the core reduces the problem to solving ordinary differential equations for that region. However, in the leading tail region, hyperbolic pdes must be solved to determine the amplitude and phase. The method of characteristics predicts triple valuedness after a caustic penumbral or cusped develops. Singular perturbation methods show the solution near first focusing satisfies the diffusion equation and involves either an incomplete Airy-type integral or an exponential integral similar to the Pearcey integral. Laplaces method shows that the critical points of the exponential phase satisfy the fundamental folding equation. A linear multi-phase solution is determined which does not become triple valued break. Instead, a wave number shock develops, which separates two different solitary wave tails, and travels at the shock velocity predicted by conservation of waves. Thus, a unique uniform leading tail solution is obtained corresponding to a specified moving core the problem is shown to be well-posed.
- Numerical Mathematics
- Plasma Physics and Magnetohydrodynamics