Related papers: Time evolution of the scattering data for a fourth…

An ordinary differential operator of the fourth order with coefficients converging at infinity sufficiently rapidly to constant limits is considered. Scattering theory for this operator is developed in terms of special solutions of the…

As a prototype of an evolution equation we consider the Schr\"odinger equation i (d/dt) \Psi(t) = H \Psi(t), H = H_0 + V(x) for the Hilbert space valued function \Psi(.) which describes the state of the system at time t in space dimension…

The topic of this paper is a semi-linear, defocusing wave equation $u_{t t}-\Delta u=-|u|^{p-1} u$ in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite energy. We prove some decay…

Scattering problem for a self-adjoint integro-differential operator, which is the sum of the operator of second derivative and of a finite-dimensional self-adjoint operator, is studied. Jost solutions are found and it is shown that the…

We discuss differential-- versus integral--equation based methods describing out--of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach,…

The transfer matrix ${\mathbf{M}}$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical…

We show that the method of splitting the operator ${\rm e}^{\epsilon(T+V)}$ to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schr\"odinger equation. These algorithms require…

Inverse scattering problem for an operator, which is a sum of the operator of the third derivative and of an operator of multiplication by a real function, is solved. The main closed system of equations of inverse problem is obtained. This…

We consider non linear elliptic equations of the form $\Delta u = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a…

We analyse the scattering operator associated with the defocusing nonlinear Schr{\"o}dinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature…

This paper is concerned with nonlinear Schr\"odinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a…

We present an iterative algorithm to compute numerical approximations of the potential for the Schr\"odinger operator from scattering data. Four different types of scattering data are used as follows: fixed energy, fixed incident angle,…

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schr\"odinger equation with a subcritical dissipative nonlinearity $\lambda |u|^\alpha u$, where $0<\alpha<2$, and $\lambda $ is a complex constant…

Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system…

In this work, we consider the inverse scattering transform and multi-solition solutions of the sextic nonlinear Schr\"{o}dinger equation. The Jost functions of spectrum problem are derived directly, and the scattering data with $t=0$ are…

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…

We prove small data scattering for the fourth-order Schr\"odinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+\Delta^2 u+\alpha u^2 + \beta \bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $\alpha, \beta…

The initial boundary value problem on a half-line for the KdV equation with the boundary conditions $u|_{x=0}=a\leq0$, $u_{xx}|_{x=0}=3a^2$ is integrated by means of the inverse scattering method. In order to find the time evolution of the…

A non-linear functional $Q[u,v]$ is given that governs the loss, respectively gain, of (doubly degenerate) eigenvalues of fourth order differential operators $L = \partial^4 + \partial u \partial + v$ on the line. Apart from factorizing $L$…

We consider a class of nonlinear Schr\"odinger equations with potential \[ i\partial_t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \] where $\frac{4}{3}<\alpha<4$ and $V$ is a Kato-type potential. We…