On nonlinear equation of Schrödinger type

Soltanov, Kamal N.

doi:10.1063/1.4765598

Cited by 1 publication

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“…It should be noted that here we continue the investigation of the problem studied in the article [31] where this problem is studied in the case Q ≡ Q T ≡ (0, T ) × Ω, when T < ∞ is some number. Here we will study the global existence of this problem, i.e.…”

Section: Proof Of Existence Theorem Of Problem (01)-(02)mentioning

confidence: 99%

“…It should be noted that the steady-state case of the problem of such type were studied in [28]. Here we study problem (0.1)-(02) globally in the dynamical case, which in [31] is studied for t ∈ (0, T ), T < ∞. Here an existence theorem (section 1) for the problem (0.1) -(0.2) is proved in the model case when f (x, u) only has the above expression (section 4).…”

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confidence: 99%

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Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation

Soltanov

2015

Ukr Math J

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We study a global mixed problem for the nonlinear Schrödinger equation with a nonlinear term in which the coefficient is a generalized function. A global solvability theorem for the analyzed problem is proved by using the general solvability theorem from [K. N. Soltanov, Nonlin. Anal.: Theory, Meth., Appl., 72, No. 1 (2010)]. We also investigate the behavior of the solution of the problem under consideration.We consider the following problem for the nonhomogeneous nonlinear Schrödinger equationwhere h(t, x) and u 0 (x) are complex functions, f (x, ⌧ ) is a distribution (generalized function) with respect to variable x 2 ⌦, ⌦ is a bounded domain with sufficiently smooth boundary @⌦, and i ⌘ p −1. We study this problem in the case where the function f (x, t) can be represented in the formi.e., the function f has the growth with respect to an unknown function of polynomial type; here, a : ⌦ −! R is a function and q : ⌦ −! R is a generalized function, p ≥ 2, e p ≥ 2, h 2 L 2 (Q) (i.e., h(t, x) ⌘ h 1 (t, x)+ih 2 (t, x) and h j 2 L 2 (Q), j = 1, 2).The nonlinear Schrödinger equation of the form (0.1) and also steady-state case of Eq. (0.1) appear in several models of different physical phenomena corresponding to various functions f. Equations of this type were studied in numerous articles under different conditions imposed on the function f both in the dynamic case (see, e.g., [2, 4, 6-9, 14, 17, 18, 20, 22, 24, 32, 33, 35] and the references therein) and in the steady-state case (see, e.g., [1, 3, 5, 10-16, 18, 19, 23-26, 28, 33, 34, 36] and the references therein). It is known that, in the steady-state case (i.e., if u is independent of t), Eq. (0.1) is an equation of the semiclassical nonlinear Schrödinger type (i.e., NLS) (see [1,2,3,10,13] and the references therein). In recent years, significant attention is given to problem (0.1) with small " > 0 as the coefficients of the linear part because the solutions are known (as in the semiclassical states), which can be used to describe the transition from quantum to classical mechanics (see [3, 10-14, 23-25, 33-36] and the references therein).In the above-mentioned papers, Eq. (0.1) and the steady-state case were considered for various functions f (x, u) that are, for the most part, Carathéodory functions 2 with some additional properties. Moreover, in some 1 Hacettepe University, Ankara, Turkey. 2 Let f : ⌦ ⇥ R m −! R be a given function, where ⌦ is a nonempty measurable set in R n and n, m ≥ 1. Then f is a Carathéodory function whenever x −! f (x, ⌘) is measurable on ⌦ for all ⌘ 2 R m and ⌘ −! f (x, ⌘) is continuous on R m for almost all x 2 ⌦ .

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Section: Proof Of Existence Theorem Of Problem (01)-(02)mentioning

confidence: 99%

mentioning

confidence: 99%