Slowly varying solitary waves. II. Nonlinear Schrödinger equation

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient nonlinear Schrodinger equation. A multiple scale method is used to determine the amplitude and phases of the wave to the second order in the perturbation parameter. The method is similar to that used in (I) (R. Grimshaw 1979 Proc. R. Soc. Lond. A 368, 359). The results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.