Related papers: On a complex differential Riccati equation

For the Riesz fractional derivative besides the well known integral representation two new differential representations are presented, which emphasize the local aspects of a fractional derivative. The consequences for a valid solution of…

Given a solution of a semilinear dispersive partial differential equation with a real analytic nonlinearity, we relate its Cauchy data at two different times by nonlinear representation formulas in terms of convergent series. These series…

Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of…

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\''{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect…

This is the second part of a two-paper series studying the nonlinear Schr\"odinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schr\"odinger equation.…

In this paper, regularity properties of Cauchy problem for linear and nonlinear abstract Schr\"odinger equations in vector-valued function spaces are obtained.

We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…

Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the…

We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional…

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are…

The notion of pairable functions is introduced and some of its properties are developed. In this connection the famous Euler identity is interpreted as a property of certain pairable functions and finite cyclic groups.

We introduce the notion of Differential Sequences of ordinary differential equations. This is motivated by related studies based on evolution partial differential equations. We discuss the Riccati Sequence in terms of symmetry analysis,…

Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…

The partial fraction expansion of coth($\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet…

Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Several versions of fractional variational…

This paper is concerned with the Cauchy-Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we…

We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties…

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary…

General features of nonlinear quantum mechanics are discussed in the context of applications to two-level atoms.

We give a connection between the Picard type theorem of Polya-Saxer-Milliox and characterization of entire solutions of a differential equation and then their higher dimensional extensions, which leads further results on both (ordinary and…