Regularity properties of Schrödinger equations in vector-valued spaces and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 149-166
Author(s):  
Veli Shakhmurov
Keyword(s):  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
The Cauchy Problem ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

Abstract In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A, we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems.

Regularity properties of nonlinear abstract Schrödinger equations and applications

2020 ◽  
Vol 31 (13) ◽  
pp. 2050105
Author(s):  
Veli Shakhmurov
Keyword(s):  
Banach Space ◽  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator [Formula: see text] defined in a Banach space [Formula: see text], in which by choosing [Formula: see text] and [Formula: see text] we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.

scholarly journals The regularity properties and blow-up of the solutions for improved Boussinesq equations

2021 ◽  
pp. 1-21
Author(s):  
Veli Shakhmurov ◽  
Rishad Shahmurov
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Blow Up ◽  
Boussinesq Equations ◽  
Qualitative Properties ◽  
Physical Systems ◽  
Regularity Properties ◽  
The Cauchy Problem ◽  
Linear And Nonlinear ◽  
The Given

In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations that include the general differential operators. First, by virtue of the Fourier multipliers, embedding theorems in Sobolev and Besov spaces, the existence, uniqueness, and regularity properties of the solution of the Cauchy problem for the corresponding linear equation are established. Here, L p -estimates for a~solution with respect to space variables are obtained uniformly in time depending on the given data functions. Then, the estimates for the solution of linearized equation and perturbation of operators can be used to obtain the existence, uniqueness, regularity properties, and blow-up of solution at the finite time of the Cauchy for nonlinear for same classes of Boussinesq equations. Here, the existence, uniqueness, L p -regularity, and blow-up properties of the solution of the Cauchy problem for Boussinesq equations with differential operators coefficients are handled associated with the growth nature of symbols of these differential operators and their interrelationships. We can obtain the existence, uniqueness, and qualitative properties of different classes of improved Boussinesq equations by choosing the given differential operators, which occur in a wide variety of physical systems.

scholarly journals Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Nakao Hayashi ◽  
Chunhua Li ◽  
Pavel I. Naumkin
Keyword(s):  
Lower Bound ◽  
Positive Root ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Time Decay ◽  
Large Initial Data ◽  
Initial Value ◽  
Optical Fields ◽  
Gauge Invariant ◽  
Dissipative Condition

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearityλup-1uof orderpn<p≤1+2/nfor arbitrarily large initial data, where the lower boundpnis a positive root ofn+2p2-6p-n=0forn≥2andp1=1+2forn=1.Our purpose is to extend the previous results for higher space dimensions concerningL2-time decay and to improve the lower bound ofpunder the same dissipative condition onλ∈C:Im⁡ λ<0andIm⁡ λ>p-1/2pRe λas in the previous works.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

Short Time Behavior of Solutions to Linear and Nonlinear Schrödinger Equations

2008 ◽  
Vol 60 (5) ◽  
pp. 1168-1200 ◽  
Author(s):  
Michael Taylor
Keyword(s):  
Broad Class ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Time Behavior ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Pinsky Phenomenon ◽  
Nonlinear Schrodinger Equations ◽  
Linear And Nonlinear ◽  
Short Time

AbstractWe examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schrödinger equations ut = iΔu+q(u) on I×ℝn, with initial data u(0, x) = f (x). Particular attention is paid to cases where f is piecewise smooth, with jump across an (n−1)-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general n in the linear case. We also have detailed analyses for a broad class of nonlinear equations when n = 1 and 2, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation.

Application of the Homotopy Perturbation Method to Linear and Nonlinear Schrödinger Equations

2008 ◽  
Vol 63 (3-4) ◽  
pp. 140-144 ◽  
Author(s):  
Mohamed M. Mousaa ◽  
Shahwar F. Ragab
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Analytic Treatment ◽  
Nonlinear Schrödinger ◽  
Homotopy Perturbation ◽  
Nonlinear Schrodinger Equations ◽  
Linear And Nonlinear

He’s homotopy perturbation method (HPM) is applied to linear and nonlinear Schrödinger equations for obtaining exact solutions. The HPM is used for an analytic treatment of these equations. The results reveal that the HPM is very effective, convenient and quite accurate to such types of partial differential equations.

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