Global solutions and ground states of a nonlinear Schrödinger equation in matrix geometry

2019 ◽  
Vol 573 ◽  
pp. 1-11
Author(s):  
Jiaojiao Li ◽  
Li Ma
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

scholarly journals The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

1991 ◽  
Vol 117 (3-4) ◽  
pp. 251-273 ◽  
Author(s):  
Thierry Cazenave ◽  
Fred B. Weissler
Keyword(s):  
Schrödinger Equation ◽  
Global Solution ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Structure Of Solutions ◽  
Nonlinear Schrödinger ◽  
Conformally Invariant

SynopsisWe study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

scholarly journals Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 617
Author(s):  
Riccardo Adami ◽  
Filippo Boni ◽  
Alice Ruighi
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States ◽  
Matching Condition ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff’s), is non-trivially interacting. This category includes Dirac’s delta conditions, delta prime, Fülöp-Tsutsui, and others.

scholarly journals Ground states of dispersion-managed nonlinear Schrödinger equation

2000 ◽  
Vol 62 (5) ◽  
pp. 7358-7364 ◽  
Author(s):  
Vadim Zharnitsky ◽  
Emmanuel Grenier ◽  
Sergei K. Turitsyn ◽  
Christopher K. R. T. Jones ◽  
Jan S. Hesthaven
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger
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