caa a22 4500 467952426 CHVBK 20180405193743.0 cr unu---uuuuu 170328e20050201xx s 000 0 eng 10.1007/s10958-005-0091-5 doi (NATIONALLICENCE)springer-10.1007/s10958-005-0091-5 (NATIONALLICENCE)springer-10.1007/s10958-005-0005-6 Novokshenov V. Mat. Inst., VTs, Ufa aut Temporal asymptotics for soliton equations in problems with step initial conditions [Elektronische Daten] [V. Novokshenov] In this survey, we present modern approaches to the construction and justification of large time asymptotics for solutions of main soliton equations with step-like initial condition whose boundary conditions as x → ±∞ are finite-gap, quasi-periodic solutions. The principal term of the asymptotic is also a finite-gap, quasi-periodic solution whose phase vectors are modulated with respect to the slow space-like variable. The Whitham equations describing this modulation are studied in detail. For the KdV equations, we construct and justify the principal term of the asymptotic for arbitrary finite-gap boundary conditions. By examining the sine-Gordon equation, we study the case of boundary conditions with complex-valued, self-conjugated quasi-periods. We prove the existence and uniqueness theorems in the case of the complex-valued group velocities that appear in the Whitham equations in the case considered. We present a complete picture (uniform in x) of the Whitham deformation for the case of 1-gap boundary conditions in the sine-Gordon equation. Springer Science+Business Media, Inc., 2005 Journal of Mathematical Sciences Kluwer Academic Publishers-Consultants Bureau 125/5(2005-02-01), 717-749 1072-3374 125:5<717 2005 125 10958 https://doi.org/10.1007/s10958-005-0091-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-005-0091-5 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Novokshenov V. Mat. Inst., VTs, Ufa aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Kluwer Academic Publishers-Consultants Bureau 125/5(2005-02-01), 717-749 1072-3374 125:5<717 2005 125 10958 NATIONALLICENCE 100 1- Novokshenov V. Mat.Inst.,VTs, Ufa aut NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-005-0005-6 text/html Onlinezugriff via DOI Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE NATIONALLICENCE NL-springer