An Exact Solution of Nonlinear Schrödinger Equation in a Lossy Fiber System Using Direct Solution Method

We present an exact solution of the nonlinear Schrödinger equation (NLSE) for beam propagation in nonlinear fiber optics. It is a lossy fiber system with the beam as solitons. Fiber losses are understood to reduce the peak power of solitons along the fiber length. That is due to its value depending on the fiber attenuation constant of α. Considering fiber loss features on the equation, we write one set modification of the NLSE and make models the main topic of our work. We solved the model and found a straightforward analytical solution of modified NLSE for the system via the direct solution method. To the best of our knowledge, no literature has presented such as solution yet. By substituting them into equations, we validate solutions. It is valid as an exact solution to the NLSE. Lastly, we found a solution offering soliton propagation suitable for the system under study

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

Second harmonic Hamiltonian: Algebraic and Schrödinger approaches

We study in detail the behavior of the energy spectrum for the second harmonic generation (SHG) and a family of corresponding quasi-exactly solvable Schrödinger potentials labeled by a real parameter b. The eigenvalues of this system are obtained by the polynomial deformation of the Lie algebra representation space. We have found the bi-confluent Heun equation (BHE) corresponding to this system in a differential realization approach, by making use of the symmetries. By means of a b-transformation from this second-order equation to a Schrödinger one, we have found a family of quasi-exactly solvable potentials. For each invariant n-dimensional subspace of the second harmonic generation, there are either n potentials, each with one known solution, or one potential with n-known solutions. Well-known potentials like a sextic oscillator or that of a quantum dot appear among them