All systems of (n+1)-dimensional quasilinear evolutional secondorder equations invariant under the chain of algebras AG(1.n) ⊂ AG1(1.n) ⊂ AG2(1.n) are described. The obtained results are illustrated by examples of nonlinear Schrödinger equations.

Witten's non-relativistic formalism of supersymmetric quantum mechanics was based on a factorization and partnership between Schrödinger equations. We show how it accommodates a transition to the partnership between relativistic Klein-Gordon equations.

We prove the existence of radially symmetric ground–states for the system of Nonlinear Schrödinger equations

1 KdV equation and Schrödinger operator 2 1.1 Integrability of Korteweg – de Vries equation . . . . . . . . . . . . . . . . . . 2 1.2 Elements of scattering theory for the Schrödinger operator . . . . . . . . . . . 5 1.3 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dressing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We investigate the dynamical behavior of continuous and discrete Schrödinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear Schrödinger counterparts. In particular, the PT-symmetric nonlinear Schrödinger equation can simultaneously support both bright and dark soliton solutions. In additi...

We suggest an effective approach to separation of variables in the Schrödinger equation with two space variables. Using it we classify inequivalent potentials V (x1, x2) such that the corresponding Schrödinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrödinger equation with the anisotropic harmonic oscillator potential V = k1x1 + k2x 2...

There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one–parameter Hamiltonian symmetries admitted by the PDEs [1]. The same result holds for semidiscrete Hamiltonian equations [2]. In this paper we consider semidiscrete canonical Hamiltonian equations. Using symmetries, we find conservation laws for...

1. Linearity and Continuity 1.1 Continuity 1.2 Linearity 1.3 Perturbation theory and linearity 1.4 Axiomatically linear equations 1.4.1 Fields, Maxwell equations 1.4.2 Densities on phase space in classical physics 1.4.3 Quantum mechanics and Schrödinger equation 2. Examples 2.1 Ordinary differential equations 2.2 The Laplace equation 2.3 The wave equation 2.4 The heat equation and Schrödinger e...

While the non-homogeneous boundary value problem for elliptic, hyperbolic and parabolic equations is relatively well understood, there are still few results for general dispersive equations. We define here a convenient class of equations comprising the Schrödinger equation, the Airy equation and linear ‘Boussinesq type’ systems, which is in some sense a generalization of strictly hyperbolic equ...