Finite time blowup for the nonlinear Schrödinger equation with a delta potential

被引：0

作者：

Hauser, Brandon ^{[1
]}

Holmes, John ^{[1
]}

O'Keefe, Eoghan ^{[1
]}

Raynor, Sarah ^{[1
]}

Yu, Chuanyang ^{[1
]}

机构：

[1] Wake Forest Univ, Dept Math & Stat, Winston Salem, NC 27109 USA

来源：

INVOLVE, A JOURNAL OF MATHEMATICS | 2023年 / 16卷 / 04期

关键词：

well-posedness; initial value problem; Schrodinger equation; NLS; Cauchy problem; Sobolev spaces;

D O I：

10.2140/involve.2023.16.591

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We study the Cauchy problem for the nonlinear Schrodinger equation with a delta potential, which can be written as iut + Au + (|u|2 sigma + c8)u = 0. We show that under certain conditions, the L infinity norm of the solution tends to infinity in finite time. In order to prove this, we study the associated Lagrangian and Hamil-tonian, and derive an estimate of the associated variance. We also derive several con-servation laws which a classical solution of the Cauchy problem must also satisfy.

引用

页码：591 / 604

页数：16

共 9 条

[1] On the Dirac delta as initial condition for nonlinear Schrodinger equations
Banica, V.
Vega, L.
[J]. ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2008, 25 (04): : 697 - 711
[2] Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data: Announcement of Results
Deift, Percy
Park, Jungwoon
[J]. LETTERS IN MATHEMATICAL PHYSICS, 2011, 96 (1-3) : 143 - 156
[3] The nonlinear Schrodinger equation on the half-line
Fokas, AS
Its, AR
Sung, LY
[J]. NONLINEARITY, 2005, 18 (04) : 1771 - 1822
[4] CLASS OF NON-LINEAR SCHRODINGER EQUATIONS .1. CAUCHY-PROBLEM, GENERAL-CASE
GINIBRE, J
VELO, G
[J]. JOURNAL OF FUNCTIONAL ANALYSIS, 1979, 32 (01) : 1 - 32
[5] Soliton splitting by external delta Potentials
Holmer, J.
Marzuola, J.
Zworski, M.
[J]. JOURNAL OF NONLINEAR SCIENCE, 2007, 17 (04) : 349 - 367
[6] DYNAMICS OF SOLITONS IN NEARLY INTEGRABLE SYSTEMS
KIVSHAR, YS
MALOMED, BA
[J]. REVIEWS OF MODERN PHYSICS, 1989, 61 (04) : 763 - 915
[7] Sulem C., 1999, The Nonlinear Schrodinger Equation: Self-Focusing and Wave Collapse
[8] Tao T., 2006, NONLINEAR DISPERSIVE, V106, DOI DOI 10.1090/CBMS/106
[9] ZAKHAROV VE, 1972, SOV PHYS JETP-USSR, V34, P62

← 1 →