2 resultados para Long wave equations

em CaltechTHESIS


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The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

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In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.