In Part I a class of linear boundary value problems is consideredwhich is a simple model of boundary layer theory. The effect of zeros andsingularities of the coefficients of the equations at the point where theboundary layer occurs is considered. The usual boundary layer techniquesare still applicable in some cases and are used to derive uniform asymptoticexpansions. In other cases it is shown that the inner and outer expansionsdo not overlap due to the presence of a turning point outside the boundarylayer. The region near the turning point is described by a two-variableexpansion. In these cases a related initial value problem is solved andthen used to show formally that for the boundary value problem either asolution exists, except for a discrete set of eigenvalues, whose asymptoticbehaviour is found, or the solution is non-unique. A proof is given of thevalidity of the two-variable expansion; in a special case this proof alsodemonstrates the validity of the inner and outer expansions.
Nonlinear dispersive wave equations which are governed by variationalprinciples are considered in Part II. It is shown that the averagedLagrangian variational principle is in fact exact. This result is used toconstruct perturbation schemes to enable higher order terms in the equationsfor the slowly varying quantities to be calculated. A simple schemeapplicable to linear or near-linear equations is first derived. Thespecific form of the first order correction terms is derived for severalexamples. The stability of constant solutions to these equations is consideredand it is shown that the correction terms lead to the instabilitycut-off found by Benjamin. A general stability criterion is given whichexplicitly demonstrates the conditions under which this cut-off occurs.The corrected set of equations are nonlinear dispersive equations and theirstationary solutions are investigated. A more sophisticated scheme isdeveloped for fully nonlinear equations by using an extension of theHamiltonian formalism recently introduced by Whitham. Finally the averagedLagrangian technique is extended to treat slowly varying multiply-periodicsolutions. The adiabatic invariants for a separable mechanical system arederived by this method.
