Dmitry Pelinovsky: PhD in Applied and Computational Mathematics

"Asymptotic Methods in Soliton Theory: Critical Problems"

Department of Mathematics, Monash University (Australia)
Advisors: R. Grimshaw, Yu.S. Kivshar
Submitted: March, 1997
Accepted: August, 1997

The thesis consists of three main parts. The first part includes the results on asymptotic derivation of nonlinear evolution equations in critical problems occuring in physics of shear stratified fluid. The second part is devoted to bifurcation methods in soliton stability theory for nonlinear evolution equations. The third part is concerned with two critical problems in the soliton theory, the critical blow-up of localized perturbations in wave systems and the structural transformation of algebraic solitary waves.

In Part I a modified asymptotic multi-scale technique is elaborated to study higher-order nonlinear effects at the dynamics of internal and interfacial waves in shallow-deep and shallow-shallow shear stratified fluids. In particular, a new, nonlocal model which is similar to the nonlinear Schrodinger equation but includes an integro-differential cubic nonlinear term is derived and found to be integrable under some special conditions. The inverse scattering transform scheme is developed to solve the initial-value problem for this equation and the Backlund--Darboux transformation, soliton solutions and infinitely many conserved quantities are also found.

In Part II a higher-order soliton perturbation theory is developed to describe adiabatic soliton dynamics induced by development of a weak instability of solitary waves in various nonlinear wave equations such as the generalized nonlinear Schodinger and Korteweg--de Vries equations. The soliton instabilities are classified into two general types according to the translational or oscillatory types of the bifurcations. The characteristic features of the translational instabilities are then considered.

In Part III asymptotic methods for derivation of blow-up rates of a critical soliton collapse are discussed and the radiation-induced effects at the singularity formation are studied for various (1+1) and (2+1) evolution equations. Besides, a new spectral theory of the algebraic soliton transformations into steady-state nonlinear waves or into pulsating wave packets is proposed for the particular example, the modified Korteweg--de Vries equation.