arXiv subjects

Juan I. Ramos

Publications and source records attributed to Juan I. Ramos.

4 recordsLinked to original sources

arXiv2013

The Nonlinear Schrödinger Equation in the Finite Line

A numerical study of the nonlinear Schrödinger (NLS) equation subject to homogeneous Dirichlet, Neumann and Robin boundary conditions in the finite line is presented. The results are compared with both the exact analytical ones for the initial-value problem (IVP) of the NLS equation and the numerical ones for periodic boundary conditions. It is shown that initial solutions obtained by truncating the exact N-soliton solution of the IVP of the NLS equation into a finite interval develop solitary waves that behave as solitons, even after collisions with the boundaries. For periodic and homogeneous Dirichlet and Neumann boundary conditions, it is observed that the interaction between solitons and boundaries is equivalent to the collision between solitons in IVP or quarterplane problems. It is shown that for homogeneous Robin boundary conditions, boundary layers that trap and delay the soliton are formed at the boundaries. Phase diagrams for the soliton amplitude at the boundary points and for the soliton's maximum amplitude show a recurrent phenomenon, and are similar to those of the cubic Duffing equation. It is also shown that the phase diagrams are strong functions of the parameter that defines the Robin boundary conditions. A method of images, similar to the one used in potential theory, is developed for the NLS equation in the quarterplane with homogeneous Dirichlet and Neumann boundary conditions at the finite boundary.

nlin.PS↗

arXiv2013

Forces on Solitons in Finite, Nonlinear, Planar Waveguides

The forces acting on and the energies of solitons governed by the nonlinear Schrödinger equation in finite, planar waveguides with periodic and with homogeneous Dirichlet, Neumann and Robin boundary conditions are determined by means of a quantum analogy. It is shown that these densities have S-shape profiles and increase as the hardness of the boundary conditions is increased.

nlin.PS↗

arXiv2013

Classical Forces on Solitons in Finite and Infinite, Nonlinear, Planar Waveguides

Conservation equations for the mass, linear momentum and energy densities of solitons propagating in finite, infinite and periodic, nonlinear, planar waveguides and governed by the nonlinear Schrödinger equation are derived. These conservation equations are used to determine classical force densities which are compared with those derived by drawing a quantum mechanics analogy between the propagation of solitons and the motion of a quantum particle in a nonlinear potential well.

nlin.PS↗

arXiv2013

A Quantum Mechanics Analogy for the Nonlinear Schrödinger Equation in the Finite Line

A quantum mechanics analogy is used to determine the forces acting on and the energies of solitons governed by the nonlinear Schrödinger equation in finite intervals with periodic and with homogeneous Dirichlet, Neumann and Robin boundary conditions. It is shown that the energy densities remain nearly constant for periodic, while they undergo large variations for homogeneous, boundary conditions. The largest variations in the force and energy densities occur for the Neumann boundary conditions, but, for all the boundary conditions considered in this paper, the magnitudes of these forces and energies recover their values prior to the interaction of the soliton with the boundary, after the soliton rebound process is completed. It is also shown that the quantum momentum changes sign but recovers its original value after the collision of the soliton with the boundaries. The asymmetry of the Robin boundary conditions shows different dynamic behaviour at the left and right boundaries of the finite interval.

nlin.PS↗