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- Title
Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system.
- Authors
Martel, Yvan; Nguyến, Tiễn Vinh
- Abstract
We consider a system of coupled cubic Schrödinger equations in one space dimension {i∂tu + ∂2xu + (|u|2 + ω|v|2)u = 0 i∂tv + ∂2xv + (|v|2 + ω|u|2)v = 0 (t,x)∈ R × R, in the non-integrable case 0 < ω < 1. First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, i.e. a solution of the system satisfying lim t → + ∞|| (u(t)v(t)) − (eitQ(⋅ − ½ log(Ωt) − ¼ log log t) eitQ (⋅ + 1½log(Ωt) + ¼ log log t)|| H1 × H1 = 0 where Q = √2 sech is the explicit solution of Q'' − Q + Q3 = 0 and Ω > 0 is a constant. This result extends to the non-integrable case the existence of symmetric 2-solitons with logarithmic distance known in the integrable case ω = 0 and ω = 1 ([15,33]). Such strongly interacting symmetric 2-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schrödinger equation in any space dimension and for any energy-subcritical power nonlinearity ([20,22]).Second, under the conditions 0 < c < 1 and 0 < ω < ½c (c+1), we construct solutions of the system satisfying lim t → + ∞ || (u(t)v(t)) − (eic2tQc (⋅ − 1(c+1)c log(Ωct)) eitQ (⋅ + 1c+1 log(Ωct) H1 × H1 = 0 where Qc(x) = cQ(cx) and Ωc > 0 is a constant. Such logarithmic regime with non-symmetric solitons does not exist in the integrable cases ω = 0 and ω = 1 and is still unknown in the non-integrable scalar case.
- Subjects
NONLINEAR Schrodinger equation; SOLITONS; CUBIC equations; SCHRODINGER equation; DISTANCES
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2020, Vol 40, Issue 3, p1595
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2020087