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with and f > 0 throughout the given domain, has a turning point
at x = 0. Find the equation that, on an appropriate fast scale (X ), describes
the solution near x = 0 (as Show that this equation has solutions
which can be written in terms of Bessel functions: for suitable
and v.
Q4.21 Schrödinger’s equation for high energy. The time-independent, one-dimensional
Schrödinger equation for a simple-harmonic-oscillator potential can be
written

where E (= constant) is the total energy. Let us write and
define to give

this equation has turning points at and we require exponentially de-
caying solutions as (and oscillatory solutions exist for
Find the leading term in each of the regions, match (and thus develop ap-
propriate connection formulae) and show that the eigenvalues (E) satisfy
where n is a large integer. (This problem can be solved
exactly, using Hermite functions; it turns out that our asymptotic evaluation of
E is exact for all n.)
Q4.22 A weakly nonlinear wave. A wave is described by the equation

where Introduce
and derive the equations that completely describe the leading-order solution

which is uniformly valid. (Do not solve your equation for
Q4.23 A weakly nonlinear wave: NLS I. A wave is described by the equation

with Introduce (where c (k)
and are independent of and seek a solution