Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential

2016 ◽  
Vol 57 (5) ◽  
pp. 051501 ◽  
Author(s):  
I. P. Naumkin
Keyword(s):  
Asymptotic Behavior ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

scholarly journals SHARP ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR SCHRÖDINGER EQUATIONS WITH REPULSIVE INTERACTIONS

2005 ◽  
Vol 07 (02) ◽  
pp. 167-176 ◽  
Author(s):  
NAOYASU KITA ◽  
TOHRU OZAWA
Keyword(s):  
Large Time Behavior ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Repulsive Interactions ◽  
The Cauchy Problem

A detailed description is given on the large time behavior of scattering solutions to the Cauchy problem for nonlinear Schrödinger equations with repulsive interactions in the short-range case without smallness condition on the data.

WAVE OPERATORS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH A NONLINEARITY OF LOW DEGREE IN ONE OR TWO SPACE DIMENSIONS

2003 ◽  
Vol 05 (06) ◽  
pp. 983-996 ◽  
Author(s):  
KAZUNORI MORIYAMA ◽  
SATOSHI TONEGAWA ◽  
YOSHIO TSUTSUMI
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Final States ◽  
Schrödinger Equations ◽  
Wave Operators ◽  
Nonlinear Schrödinger ◽  
Low Degree ◽  
Nonlinear Schrodinger Equations ◽  
Two Space Dimensions

In this paper, we study the asymptotic behavior of solutions to the cubic and the quadratic nonlinear Schrödinger equations in one and two space dimensions, respectively. When the nonlinearity is of a form f = |u|p-1u, it is known that there exist scattered states if p > 1+2/n and there does not otherwise. Therefore we may consider the nonlinearities treated in the present paper to be of critical order for the existence of scattered states though their forms differ slightly from that given above. We prove, however, that there exist scattered states for these critical nonlinear Schrödinger equations, in other words, that the wave operators exist on a certain set of final states. Our proof is mainly based on the construction of suitable approximate functions that approach to solutions of nonlinear Schrödinger equations at t = ∞.

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

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