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From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schr\"odinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly…
We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows…
This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a…
A variationally improved Sturmian approximation for solving time-independent Schr\"odinger equation is developed. This approximation is used to obtain the energy levels of a quartic anharmonic oscillator, a quartic potential, and a Gaussian…
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated…
We perform the complete group classification in the class of nonlinear Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+|\psi|^\gamma\psi+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$…
This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of…
A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems. It is assumed that constraint function values and derivatives are available, but only stochastic…
The aim of this paper is to find the exact solutions of the Schrodinger equation. As is known, the Schrodinger equation can be reduced to the continuum equation. In this paper, using the non-linear Legendre transform the equation of…
A method of representation of a solution as segments of the series in powers of the step of the independent variable is expanded for solving complex systems of ordinary differential equations (ODE): the Lorenz system and other systems. A…
Alternative machine learning approaches that are computationally light with low latency and can work with only a small training dataset are needed for applications where the insatiable demand of deep learning methods for computing power and…
We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schr\"odinger equation. The generalization yields numerical solutions accurate to order $(\Delta x)^{2r-1}$ in…
We present a novel numerical method and algorithm for the solution of the 3D axially symmetric time-dependent Schr\"odinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent…
We develop an approach to solving numerically the time-dependent Schrodinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an…
The most common method for calculating accurate numerical solutions for complicated linear differential equations - for example, finding eigenvalues and eigenfunctions of the Schrodinger equation for many-electron atoms - is the variational…
We introduce two multiscale numerical schemes for the time integration of weakly nonlinear Schr\"odinger equations, built upon the discretization of Picard iterates of the solution. These high-order schemes are designed to achieve high…
We demonstrate that it is possible to compute wave function normalization constants for a class of Schr\"odinger type equations by an algorithm which scales linearly (in the number of eigenfunction evaluations) with the desired precision P…
In this paper, the trial function method is employed to find the exact solutions for high-order nonlinear Schr\"odinger equations with time-dependent coefficients. This system describes the propagation of ultrashort light pulses in…
We present a new method for the solution of the Schrodinger equation applicable to problems of non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: An…