An Integral Form of the Nonlinear Schroedinger Equation with Variable Coefficients (original) (raw)

Schrödinger Equations in Nonlinear Systems

2019

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On Integrability of Nonautonomous Nonlinear Schroedinger Equations

arXiv (Cornell University), 2010

We show, in general, how to transform the nonautonomous nonlinear Schrödinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is also outlined. A few simple integrable systems are discussed.

On the solution of multicomponent nonlinear Schrödinger equations

Physics Letters A, 2004

Using algebraic approach we find new solutions of N-component nonlinear Schrödinger equations with mixed signs of nonlinear coefficients. Explicit expressions are given for the cases N = 2 and N = 3. These solutions admit singularities for certain fixed values of the spatial variable.

A Higher Order Nonlinear Schrödinger Equation

Jurnal Riset dan Aplikasi Matematika (JRAM), 2020

Nonlinear Schrödinger (NLS) equation has been widely studied, and it has been appeared in tremendous amount of papers. NLS equation models a wave packet travelling in dispersive and nonlinear media. In this paper, a higher order NLS equation is discussed. The solution, which is complex wave envelope, is investigated numerically for narrow and broad envelope. Broader envelope shows deformation during the evolution, while narrow envelope does not. Another finding is that the fifth order nonlinearity does not contribute significantly to the envelope deformation. Hence, working with higher order will take much effort but insignificant results.

Solution of the Cauchy problem for a time-dependent Schr�dinger equation

J Math Phys Ny, 2008

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product. 2 (1.6) Date: May 26, 2018. 1991 Mathematics Subject Classification. Primary 81Q05, 33D45, 35C05, 42A38; Secondary 81Q15, 20C35. Key words and phrases. The Cauchy initial value problem, the Schrödinger equation, Hamiltonian, harmonic oscillator, the hypergeometric function, hyperspherical harmonics, Bargmann function, the Laguerre polynomials, the Meixner polynomials, the Meixner-Pollaczek polynomials.

Solution of the Cauchy problem for a time-dependent Schrödinger equation

Journal of Mathematical Physics, 2008

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product.

On integrability of nonautonomous nonlinear Schrödinger equations

Proceedings of the American Mathematical Society, 2012

We show, in general, how to transform the nonautonomous nonlinear Schrödinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is also outlined.

Some new exact Solutions for the nonlinear schrödinger equation

In this paper, we apply the generalized Bernoulli sub-ODE method to seek exact solutions for nonlinear evolution equations arsing in the theory of mathematical physics. For testing the validity of this method, we use it to derive exact solutions for the nonlinear Schrödinger (NLS) equation. As a result, some exact solutions for it are successfully found with the aid of the mathematical software Maple.

On uniqueness of solution for a nonlinear Schrödinger–Airy equation

Nonlinear Analysis: Theory, Methods & Applications, 2006

We prove that a sufficiently smooth solution to the initial value problem associated to the equation j t u + i j 2 x u + j 3 x u + i |u| 2 u + |u| 2 j x u + u 2 j x u = 0, x,t ∈ R, is uniquely determined by its values in the semi-line at two different instants of time.