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
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.