Perturbations of the KdV and MKdV equation, of the sine-Gordon equation, of a nonlinear Schrodinger equation, of the modified KdV equation, and of the Phi-4 equation are studied under the assumption that the nonperturbed problem has a one-soliton solution. It is proved that the parameters of the perturbed solution are defined uniquely and do not depend on the ambiguities that remain in the corrections if these corrections do not contain secular terms.

Weak perturbation of the KdV equation with multisoliton initial data is studied. The ordinary differential equations for the slow modulation of the soliton parameters are derived. It is found that the phase shift of each soliton depends on the slowly varying amplitudes of all fore solitons. The first correction of the asymptotics is obtained. The leading term outside of the soliton sectors is given, which is called the soliton tail.

A new method for analyzing the problems of chemical kinetics is elaborated involving the technique of mathematical modeling. Namely, the matching method of the asymptotic expansion is applied to analyzing the inhibition mechanism of oxidation. The proposed approach is an extension of the well-known method of quasi-stationary concentrations and may be applied to study a series of problems in the field of chemical kinetics. Three different time scales were established for the mechanism of inhibited oxidation I -->i r, r + RH --> rH + R, R + O2 -->k1 RO2, RO2 + RH -->k2 ROOH + R, RO2 + RO2 -->k6 P6, RO2 + InH -->k7 ROOH + In, RO2 + In -->k8 P8, In + In -->k9 P9 under restrictions k7[InH]0/(2k6W(i))1/2 less-than-or-equal-to 1 and k8 >> 2k6 >> k7. At the first time scale (that is very fast and is measured in second fractions) the concentration of radicals In only changes while [RO2] congruent-to [RO2]0, [InH] congruent-to [InH]0 are constants. At the second time scale(s), [RO2] changes while [In] congruent-to [In]st, [InH] congruent-to [InH]0 are constants. At the third time scale (min), [In H] changes. An asymptotic analysis of the differential equations allows us to find out both the time duration of each step and the variation of the component which changes at this step. After that the rate constants k8, 2k6, k7 are determined from comparison with the experimental measurements of [In], [RO2], and [In H]. Due to the simplicity and efficiency of the asymptotic method, one may be applied to treating the complex multicenter radical chain processes such as conjugated oxidation, radical copolymerization, sulfoxidation, etc. (C) 1993 John Wiley & Sons, Inc.

CONTENTS Introduction § 1. Hyperbolic problems without dispersion. Asymptotic decomposition of a small amplitude solution into simple waves § 2. The method of averaging in hyperbolic problems without dispersion § 3. The method of matching in hyperbolic problems without dispersion § 4. Long waves in problems with weak dissipation § 5. Long waves in problems with small dispersion § 6. Small amplitude solutions in problems with strong dispersion § 7. Continuum limits of discrete equations on small amplitude solutions § 8. Asymptotic passages in non-one-dimensional waves § 9. Comments, problems References

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