Painlevé analysis of the damped, driven nonlinear Schrödinger equation

1988 ◽  
Vol 109 (1-2) ◽  
pp. 109-126 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Analytic Functions ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Completely Integrable ◽  
Real Analytic Functions ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equationwhere a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only ifwhere α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

2000 ◽  
Vol 130 (5) ◽  
pp. 1029-1043 ◽  
Author(s):  
N. Hayashi ◽  
E. I. Kaikina ◽  
P. I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Nonlinear Schrodinger Equation ◽  
Time Decay ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Image Position

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrödinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

ON A CLASS OF NONLINEAR SCHRÖDINGER EQUATIONS ON FINITE GRAPHS

2020 ◽  
Vol 101 (3) ◽  
pp. 477-487
Author(s):  
SHOUDONG MAN
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Integral Inequality ◽  
Schrodinger Equation ◽  
Type Inequality ◽  
Finite Graph ◽  
Graph Laplacian ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Image Position

Suppose that $G=(V,E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\unicode[STIX]{x1D6E5}$ be the usual graph Laplacian. Consider the nonlinear Schrödinger equation of the form $$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V),\end{eqnarray}$$ on the graph $G$, where $f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$ is a nonlinear real-valued function and $\unicode[STIX]{x1D6FC}$ is a parameter. We prove an integral inequality on $G$ under the assumption that $G$ satisfies the curvature-dimension type inequality $CD(m,\unicode[STIX]{x1D709})$. Then by using the Poincaré–Sobolev inequality, the Trudinger–Moser inequality and the integral inequality on $G$, we prove that there is a nontrivial solution to the nonlinear Schrödinger equation if $\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709})$, where $\unicode[STIX]{x1D706}_{1}$ is the first positive eigenvalue of the graph Laplacian.

A symmetry-breaking phenomenon and asymptotic profiles of least-energy solutions to a nonlinear Schrödinger equation

2005 ◽  
Vol 135 (2) ◽  
pp. 357-392 ◽  
Author(s):  
Kazuhiro Kurata ◽  
Tatsuya Watanabe ◽  
Masataka Shibata
Keyword(s):  
Schrödinger Equation ◽  
Symmetry Breaking ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Least Energy Solution ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

In this paper, we study a symmetry-breaking phenomenon of a least-energy solution to a nonlinear Schrödinger equation under suitable assumptions on V(x), where λ > 1, p > 2 and χA is the characteristic function of the set A = [−(l + 2), −l] ∪ [l,l + 2] with l > 0. We also study asymptotic profiles of least-energy solutions for the singularly perturbed problem for small ε > 0.

scholarly journals Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

2012 ◽  
Vol 142 (6) ◽  
pp. 1237-1262 ◽  
Author(s):  
Patricio Felmer ◽  
Alexander Quaas ◽  
Jinggang Tan
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Symmetry Properties ◽  
Existence Of Positive Solutions ◽  
Image Position

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

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