On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

Mohamed Jleli,Calogero Vetro,Bessem Samet

doi:10.3390/sym12071197

Mohamed Jleli, Calogero Vetro + Show 1 more

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https://doi.org/10.3390/sym12071197

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Journal: Symmetry	Publication Date: Jul 20, 2020
Citations: 4	License type: CC BY 4.0

Affiliation: King Saud University, University of Palermo

Abstract

This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form i α ∂ t α ω ( t , z ) + a 1 ( t ) Δ ω ( t , z ) + i α a 2 ( t ) ω ( t , z ) = ξ | ω ( t , z ) | p , ( t , z ) ∈ ( 0 , ∞ ) × R N , where N ≥ 1 , ξ ∈ C \ { 0 } and p > 1 , under suitable initial data. To establish our nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefficient functions a 1 , a 2 ∈ L l o c 1 ( [ 0 , ∞ ) , R ) , and provide two illustrative examples.

Highlights

Introduction and PreliminariesIn this paper, we study the following initial value problem for the fractional in time nonlinearSchrödinger equation iα ∂αt ω (t, z) + a1 (t)∆ω (t, z) + iα a2 (t)ω (t, z) = ξ |ω (t, z)| p, (t, z) ∈ (0, ∞) × R N, (1)where N ≥ 1, ξ ∈ C\{0} and p > 1, under the assumption that ω (0, z) = v (z), z ∈ R N, N ≥ 1. (2) απIn the left-hand side of (1), iα = ei 2, and ∂αt means the Caputo fractional derivative in time of order α ∈ (0, 1)
We need some regularities of the coefficient functions: a1, a2 ∈ L1loc ([0, ∞), R), a1 6≡ 0, and v ∈ L1loc (R N, C)
The interest for nonlinear Schrödinger equations is dictated by various applications in physics

Summary

Introduction and Preliminaries

We study the following initial value problem for the fractional in time nonlinear. A significant topic for nonlinear partial Schrödinger equations is to establish sufficient conditions for the existence of solutions providing a localized behavior. The authors considered the global behavior of solutions to problem (3), they established a finite-time blow-up result of an L2 -solution whenever p ∈ (1, 1 + N2 ). For further interesting contributions to the analysis of the blow-up behavior of solutions to fractional nonlinear Schrödinger problems, we mention the papers of Fino–Dannawi–Kirane [12]. We recall the paper of Li–Ding–Xu [15] where a cubic non-polynomial spline method is implemented to solve the time-fractional nonlinear Schrödinger equation.

Non-Existence Theorem and Implications

Proof of Theorem 1

Conclusions