Existence of solutions of non-linear Schrödinger equations

1977 ◽  
Vol 76 (3) ◽  
pp. 227-232 ◽  
Author(s):  
Klaus Schmitt ◽  
Long-yi Tsai
Keyword(s):  
Periodic Potential ◽  
Differential Operators ◽  
Existence Of Solutions ◽  
Type Theorem ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Non Linear ◽  
Existence Of Periodic Solutions ◽  
Linear Differential ◽  
Intermediate Value

SynopsisThe existence of periodic solutions of non-linear Schrödinger equations with periodic potential is investigated. The main result obtained is an intermediate value type theorem for such non-linear differential operators.

Discrete differential operators for periodic and two-periodic time functions

2019 ◽  
Vol 38 (1) ◽  
pp. 325-347 ◽  
Author(s):  
Tadeusz Sobczyk ◽  
Michał Radzik ◽  
Natalia Radwan-Pragłowska
Keyword(s):  
Differential Operators ◽  
High Accuracy ◽  
Large Set ◽  
Periodic Functions ◽  
Content Type ◽  
Second Derivatives ◽  
Non Linear ◽  
Discrete Operators ◽  
Linear Differential ◽  
Very High

Purpose To identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions. Design/methodology/approach The development of relations between the values of first and second derivatives of periodic and two-periodic functions, as well as the values of the functions themselves for a set of time instants. Numerical tests of discrete operators for selected periodic and two-periodic functions. Findings Novel discrete differential operators for periodic and two-periodic time functions determining their first and the second derivatives at very high accuracy basing on relatively low number of points per highest harmonic. Research limitations/implications Reduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist. Practical implications Application to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time. Originality/value Identify novel discrete differential operators for periodic and two-periodic time functions engaging a large set of time instants that determine the first and second derivatives with very high accuracy.

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