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194 4. The method of multiple scales



               where         Determine c (k) and  (and confirm that  satisfies the
               usual condition for the group speed) and find the equation for
          Q4.24 A weakly nonlinear wave: NLS II. See  Q4.23;  repeat all this for the equation




          Q4.25 A weakly nonlinear wave: NLS III. See  Q4.23;  repeat all this for the  equation




          Q4.26 Kd V   NLS. Consider the Korteweg-de Vries (KdV) equation





               where        is aparameter.Introduce                       and
                        (and here   and     will be corrections to  the original c  and
               because the given KdV equation has already been written in a suitable moving
               frame). Seek a solution






               where             find c(k),   and the equation for       (This
               example  demonstrates  that  an underlying  structure of the  KdV equation  is
               an NLS  (Nonlinear Schrödinger) equation; indeed, it can be shown that, in
               the context of water waves, for example,  the relevant NLS equation for that
               problem matches to this NLS equation—see Johnson,  1997.)
          Q4.27 Ray theory. A wave (moving in two dimensions) slowly evolves, on the scale
               so that                                          where
                             show that
                (a)              where            and
               (b)            (the eikonal  equation);
               (c)           (so the vector k is ‘irrotational’).
                (Given  that the  energy in  the  wave  motion is E(X, Y,  T), it  can be  shown
               that                     where            All this is the basis for ray
               theory, or the theory of geometrical optics, which is used to describe the properties
               of waves that move through a slowly changing environment.)
          Q4.28 Boundary-layer problem I.  Use the method of multiple scales to find,  completely,
               the first term of a uniformly valid asymptotic expansion of the solution of




               where
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