Asymptotic Methods for Wave and Quantum Problems
The collection consists of four papers in different areas of mathematical physics united by the intrinsic coherence of the asymptotic methods used. The papers describe both the known results and most recent achievements, as well as new concepts and ideas in mathematical analysis of quantum and wave problems. In the introductory paper ``Quantization and Intrinsic Dynamics'' a relationship between quantization of symplectic manifolds and nonlinear wave equations is described and discussed from the viewpoint of the weak asymptotics method (asymptotics in distributions) and the semiclassical approximation method. It also explains a hidden dynamic geometry that arises when using these methods. Three other papers discuss applications of asymptotic methods to the construction of wave-type solutions of nonlinear PDE's, to the theory of semiclassical approximation (in particular, the Whitham method) for nonlinear second-order ordinary differential equations, and to the study of the Schrodinger type equations whose potential wells are sufficiently shallow that the discrete spectrum contains precisely one point. All the papers contain detailed references and are oriented not only to specialists in asymptotic methods but also to a wider audience of researchers and graduate students working in partial differential equations and mathematical physics.
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Weak asymptotics method and interaction of nonlinear waves
Global asymptotics and quantization rules for nonlinear differential
Asymptotics of eigenfunctions in shallow potential wells and related
2ero Airy functions algebra amplitude approximate assume asymptotic expansions asymptotic solution calculate Cauchy data Cauchy problem coefficients Colombeau condition const constant construct cosh curve decreasing defined deformation quantization denote derivative determined differential equations domain dynamics eigenfunction eigenvalue English transl estimate Ether Hamiltonian exponential expression formulas free boundaries geometry global asymptotic Heaviside function Hence implies initial data integral KdV type equations kernel leading term Lemma Let us consider linear Maslov Math model equation modified Stefan problem Moreover nonlinear waves obtain Painleve equation parameter perturbation phase field system phase shift phase space Phys polynomials prove quantization rule quantum regulari2ation regularization relation respect result right-hand side satisfies Section semiclassical shock waves singular smooth functions solitary waves soliton solution of equation solution of problem solvability Stefan problem subalgebra Subsection Substituting symplectic connection symplectic manifold theory tion trajectory transform turning points variable weak asymptotics method weak solution Whitham