scholarly journals On the Existence and Stability of Standing Waves for 2-Coupled Nonlinear Fractional Schrödinger System

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Xiuyan Sha ◽  
Huanmin Ge ◽  
Jie Xin
Keyword(s):  
Minimization Problem ◽  
Standing Waves ◽  
Constrained Minimization ◽  
Stable Set ◽  
Initial Value ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Existence And Stability ◽  
Constrained Minimization Problem ◽  
Schrödinger System

We study a system of 2-coupled nonlinear fractional Schrödinger equations. Firstly, we construct constrained minimization problem to the system. Next, we prove the existence of standing waves for the system by using the concentration-compactness and commutator estimates method. Lastly, we also consider the set of minimizers of the constrained minimization problem. We prove that it is a stable set for initial value of the problem; that is, a solution to the system with initial value which is near the set will remain near it for all time.

scholarly journals Existence and Stability of Standing Waves for Nonlinear Fractional Schrödinger Equations with Hartree Type and Power Type Nonlinearities

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Na Zhang ◽  
Jie Xin
Keyword(s):  
Minimization Problem ◽  
Standing Waves ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Power Type ◽  
Initial Value ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Existence And Stability ◽  
Constrained Minimization Problem

We consider the standing wave solutions for nonlinear fractional Schrödinger equations with focusing Hartree type and power type nonlinearities. We first establish the constrained minimization problem via applying variational method. Under certain conditions, we then show the existence of standing waves. Finally, we prove that the set of minimizers for the initial value problem of this minimization problem is stable.

Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity

2016 ◽  
Vol 15 (05) ◽  
pp. 699-729 ◽  
Author(s):  
Yonggeun Cho ◽  
Mouhamed M. Fall ◽  
Hichem Hajaiej ◽  
Peter A. Markowich ◽  
Saber Trabelsi
Keyword(s):  
Cauchy Problem ◽  
Standing Waves ◽  
Orbital Stability ◽  
Schrodinger Equations ◽  
Well Posedness ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Existence And Stability

This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method.

scholarly journals A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 890
Author(s):  
Suthep Suantai ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Image Processing ◽  
Minimization Problem ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Image Inpainting ◽  
Constrained Minimization ◽  
Convex Minimization Problem ◽  
Mild Conditions ◽  
Constrained Minimization Problem ◽  
Weak Convergence Theorem

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

Finite Strain Elastostatics With Stiffening Materials: A Constrained Minimization Model

1997 ◽  
Vol 64 (2) ◽  
pp. 440-442 ◽  
Author(s):  
S. J. Hollister ◽  
J. E. Taylor ◽  
P. D. Washabaugh
Keyword(s):  
Boundary Value Problem ◽  
Minimization Problem ◽  
Finite Strain ◽  
Necessary Conditions ◽  
Piecewise Linear ◽  
Process Parameter ◽  
Constrained Minimization ◽  
Problem Statement ◽  
Stress Strain ◽  
Constrained Minimization Problem

Finite strain elastostatics is expressed for general anisotropic, piecewise linear stiffening materials, in the form of a constrained minimization problem. The corresponding boundary value problem statement is identified with the associated necessary conditions. Total strain is represented as a superposition of variationally independent constituent fields. Net stress-strain properties in the model are implicit in terms of the parameters that define the constituents. The model accommodates specification of load fields as functions of a process parameter.

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