Estimates for a slowly-varying wave equation with a periodic potential

✍ Scribed by Cathleen S. Morawetz

• Publisher: John Wiley and Sons
• Year: 1973
• Tongue: English
• Weight: 517 KB
• Volume: 26
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160260523

✦ Synopsis

Estimates for a Slowly-Varying Wave Equation with a Periodic Potential* CATHLEEN S. MORAWETZ Consider the equation for t 2 0, where P ( t ) is a smooth function of period 2 ~. I n general the solutions of the homogeneous initial value problem will grow exponentially and so too will the solutions of the non-homogeneous equation. However, with limitations on the initial data e.g., ut fixed or analytic data, one finds solutions with better growth properties. This is a first effort to study the perturbation of amplitude-modulated periodic waves, see [l], [2]. To make an analogous problem one takes g(x, t ) = G( EX, t ) . Then one wants to know the behavior of solutions at a time of the order 1 / ~. Exponential growth in time would yield solutions exploding like exp { l / ~} . Are there any which grow like a negative power of E ?

We consider then the equation with q = E X :

and note that the Fourier transform U(A, t ) of u satisfies the non-homogeneous Hill equation

(2)

ut, + ( p + P ( t ) ) U = G(A, t )

with p = E~ A2. I t is the special properties of the solutions of the homogeneous equation that may be found in [3] that are used extensively here.

A basic assumption must be made about P ( t ) to obtain the results described here :l

The origin is a n interior or right end-point of an unstable interval [po , p l ]

* The research in this paper was performed a t the Courant Institute and supported by the Office of Naval Research, under contract N00014-67A-U467-0024. Reproduction in whole or in part is permitted for any purpose of the United States Government.

For the perturbation problem of the amplitude-modulated solutions mentioned above this condition is satisfied if the initial value problem for the amplitude equations is well posed (see

[!I, [ Z ] , [4]

). There, in fact, p1 = 0.