Scattering operator for the one-dimensional Dirac equation with power nonlinearity

2016 ◽  
Vol 13 (04) ◽  
pp. 821-832
Author(s):  
Nakao Hayashi ◽  
Hironobu Sasaki
Keyword(s):  
Dirac Equation ◽  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Scattering Operator ◽  
One Dimensional ◽  
Power Nonlinearity ◽  
Nonlinear Dirac Equation ◽  
The One

We consider the nonlinear Dirac equation with a power nonlinearity [Formula: see text] where [Formula: see text], [Formula: see text]. We prove the existence of the scattering operator in the neighborhood of the origin of the weighted Sobolev space [Formula: see text], where [Formula: see text].

scholarly journals Modified Scattering for the One-Dimensional Cubic NLS with a Repulsive Delta Potential

2018 ◽  
Vol 2019 (24) ◽  
pp. 7577-7603
Author(s):  
Satoshi Masaki ◽  
Jason Murphy ◽  
Jun-Ichi Segata
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Weighted Sobolev Space ◽  
Global Solutions ◽  
Initial Value ◽  
Small Initial Data ◽  
One Dimensional ◽  
Delta Potential ◽  
The One ◽  
Cubic Nonlinear Schrödinger Equation

Abstract We consider the initial-value problem for the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that decay in $L^{\infty }$ and exhibit modified scattering.

scholarly journals Intertwining technique for the one-dimensional stationary Dirac equation

2003 ◽  
Vol 305 (2) ◽  
pp. 151-189 ◽  
Author(s):  
L.M. Nieto ◽  
A.A. Pecheritsin ◽  
Boris F. Samsonov
Keyword(s):  
Dirac Equation ◽  
One Dimensional ◽  
The One ◽  
Stationary Dirac Equation

Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems

2021 ◽  
Vol 31 (01) ◽  
pp. 2150005
Author(s):  
Ziyatkhan S. Aliyev ◽  
Nazim A. Neymatov ◽  
Humay Sh. Rzayeva
Keyword(s):  
Dirac Equation ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Bifurcation From Infinity ◽  
The One ◽  
Nodal Properties

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

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