Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schrödinger equation

Stability of nonlinear Dirac solitons under the action of external potential

The instabilities observed in direct numerical simulations of the Gross-Neveu equation under linear and harmonic potentials are studied. The Lakoba algorithm, based on the method of characteristics, is performed to numerically obtain the two spinor components. We identify non-conservation of energy and charge in simulations with instabilities, and we find that all studied solitons are numerically stable, except the low-frequency solitons oscillating in the harmonic potential over long periods of time. These instabilities, as in the case of the Gross-Neveu equation without potential, can be removed by imposing absorbing boundary conditions. The dynamics of the soliton is in perfect agreement with the prediction obtained using an Ansatz with only two collective coordinates, namely, the position and momentum of the center of mass. We employ the temporal variation of both field energy and momentum to determine the evolution equations satisfied by the collective coordinates. By applying the same methodology, we also demonstrate the spurious character of the reported instabilities in the Alexeeva-Barashenkov-Saxena model under external potentials.

Anomalous diffusion of discrete solitons driven by evolving disorder

Anomalous diffusion is simulated in this paper by studying the transport of discrete solitons in a lattice with evolving disorder. We find a Richardson-type diffusion for the small solitons and a regime of transient diffusion for larger solitons within the ensemble-averaged description. As a comparison, the time-averaged observables present a ballistic scaling for both cases. However, distribution of these observables changes remarkably with the soliton size. Our results suggest violation of ergodicity for the solitons' diffusive processes, which are expected to shed light on further understanding of the discreteness-disorder-nonlinearity interaction.

Soliton dynamics in optical fibers using the generalized traveling-wave method

The generalized traveling wave method (GTWM) is applied to the nonlinear Schrödinger (NLS) equation with general perturbations in order to obtain the equations of motion for an ansatz with six collective coordinates, namely the soliton position, the amplitude, the inverse of the soliton width, the velocity, the chirp, and the phase. The advantage of the new ansatz is that it yields three pairs of canonically conjugated coordinates and momenta that all are well-behaved. The new ansatz is applied to model the dynamics of a soliton in a dispersion-shifted optical fiber described by the generalized NLS, including dissipation, higher-order dispersion, Raman scattering, and self-steepening perturbations. It is shown that the GTWM is equivalent to the modified method of moments, which considers the time variation of the norm, the first and the second moment of the norm, the momentum, the first moment of the momentum, and the energy for the perturbed NLS equation.

Nonlinear Dirac equation solitary waves in external fields

We consider nonlinear Dirac equations (NLDE's) in the 1+1 dimension with scalar-scalar self-interaction g2/κ+1(Ψ[over ¯]Ψ)κ+1 in the presence of various external electromagnetic fields. We find exact solutions for special external fields and we study the behavior of solitary-wave solutions to the NLDE in the presence of a wide variety of fields in a variational approximation depending on collective coordinates which allows the position, width, and phase of these waves to vary in time. We find that in this approximation the position q(t) of the center of the solitary wave obeys the usual behavior of a relativistic point particle in an external field. For time-independent external fields, we find that the energy of the solitary wave is conserved but not the momentum, which becomes a function of time. We postulate that, similarly to the nonlinear Schrödinger equation (NLSE), a sufficient dynamical condition for instability to arise is that dP(t)/dq[over ̇](t)<0. Here P(t) is the momentum of the solitary wave, and q[over ̇] is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials that we always have dP(t)/dq[over ̇](t)>0, so, when instabilities do occur, they are due to a different source. We investigate the accuracy of our variational approximation using numerical simulations of the NLDE and find that, when the forcing term is small and we are in a regime where the solitary wave is stable, that the behavior of the solutions of the collective coordinate equations agrees very well with the numerical simulations. We found that the time evolution of the collective coordinates of the solitary wave in our numerical simulations, namely the position of the average charge density and the momentum of the solitary wave, provide good indicators for when the solitary wave first becomes unstable. When these variables stop being smooth functions of time (t), then the solitary wave starts to distort in shape.

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

We investigate the dynamics of traveling oscillating solitons of the cubic nonlinear Schrödinger (NLS) equation under an external spatiotemporal forcing of the form f(x,t)=aexp[iK(t)x]. For the case of time-independent forcing, a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where K(t) is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" p(v), where p(t) and v(t) are the normalized momentum and the velocity of the soliton, has a section with a negative slope. In the case of a constant K and zero damping, we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.