Abstract
We study the gain of regularity for the initial value problem for a coupled nonlinear Schrodinger system that describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photo refractive media in optics and Bose-Einstein condensates. This study is motivated by the results obtained by N. Hayashi et al.
Highlights
We study the gain of regularity for the initial value problem for a coupled nonlinear Schrodinger system that describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photo refractive media in optics and Bose-Einstein condensates
Menyuk [10,11] showed that the evolution of two orthogonal pulse envelopes in birefringent optical fiber is governed by the coupled nonlinear Schrodinger system (1.1)-(1.4)
An evolution equation enjoys a gain of regularity if their solutions are smoother for t > 0 than its initial data
Summary
We will use the following standard notation. For 1 ≤ p ≤ ∞, Lp(R) are all complex valued measurable functions on R such that |u|p is integrable for 1 ≤ p < ∞ and sup ess |u(x)| is finite for p = ∞. For a non-negative integer m and 1 ≤ p ≤ ∞, we denote by Hm(R) the Sobolev space of functions in L2(R) having all derivatives of order ≤ m belonging to L2(R). For any interval I of R and any Banach space X with the norm || · ||X , we denote by C(I : X)(respectively Cb(I : X) the space of continuous(respectively bounded continuous) functions from I to X. For an interval I, the space Lp(I : X) is the space consisting of all strongly measurable X-valued functions u(t) defined on I such that ||u||X ∈ Lp(I). Throughout this paper c is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities
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