Backlund transformation and L2-stability of NLS solitons

Physical Review E, 2015

A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p (v) < 0 is a sufficient condition for instability, while p (v) > 0 is a necessary condition for stability; here, v is the soliton velocity and p = P /N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate p(v) exactly for several classes and cases of the generalized NLSE: a soliton moving in a real potential, in particular a time-dependent ramp potential, and a time-dependent confining quadratic potential, where the nonlinearity in the NLSE also has a time-dependent coefficient. Moreover, we investigate a logarithmic and a cubic NLSE with a time-independent quadratic potential well. In the latter case, there is a bisoliton solution that consists of two solitons with asymmetric shapes, forming a bound state in which the shapes and the separation distance oscillate. Finally, we consider a cubic NLSE with parametric driving. In all cases, the p(v) curve is calculated either analytically or numerically, and the stability criterion is confirmed.

Journal of Nonlinear Science

Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov-Ma breathers and the Akhmediev breathers of the focusing nonlinear Schrödinger equation. We use the Darboux transformation to construct simultaneously the breathers and the exact solutions of the Lax system associated with the breathers. We obtain a full description of the Lax spectra for the two breathers, including multiplicities of eigenvalues. Solutions of the linearized NLS equations are then obtained from the eigenfunctions and generalized eigenfunctions of the Lax system. While we do not attempt to prove completeness of eigenfunctions, we aim to determine the entire set of solutions of the linearized NLS equations generated by the Lax system in appropriate function spaces.

arXiv (Cornell University), 2016

Physica D: Nonlinear Phenomena, 2003

We prove the global well-posedness of the one-dimensional Zakharov-Rubenchik equation iB t + ωB xx − k(u − 1 2 vρ + q|B| 2)B = 0, θρ t + (u − vρ) x = −k|B| 2 x , θu t + (βρ − vu) x = 1 2 kv|B| 2 x in the space H 2 (R) × H 1 (R) × H 1 (R). We also prove the existence and the orbital stability of solitary wave solutions to the above equations.

Progress In Variational Methods, 2010

In this paper we present some recent results concerning the existence, the stability and the dynamics of solitons occurring in the nonlinear Schroedinger equation when the parameter h → 0.

2016

We study the one-dimensional nonlinear Schrödinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound states coexist, with different powers and same propagation constant.

2008

Asymptotic stability of small solitons in one dimension is proved in the framework of a discrete nonlinear Schrodinger equation with septic and higher power-law nonlinearities and an external potential supporting a simple isolated eigenvalue. The analysis relies on the dispersive decay estimates from Pelinovsky & Stefanov (2008) and the arguments of Mizumachi (2008) for a continuous nonlinear Schrodinger equation in one dimension. Numerical simulations suggest that the actual decay rate of perturbations near the asymptotically stable solitons is higher than the one used in the analysis.

2012

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schr\&quot;odinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact) interaction with strength $\alpha$, which consists of a singular perturbation of the laplacian described by a selfadjoint operator $H_{\alpha}$, where the strength $\alpha$ depends on the wavefunction: $i\dot u= H_\alpha u$, $\alpha=\alpha(u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, we let the strength $\alpha$ depend on $u$ according to the law $\alpha=-\nu|q|^\sigma$, with $\nu &gt; 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. For such a model we prove the existence of standing waves of the form $u (t)=e^{i\omega t}\Phi_{\omega}$, which are orbitally stable in the range $\sigma \in (0,1)$, and orbitally unstable for $\sigma \geq 1.$...

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009

We consider splitting and stabilization of second-order solitons (2-soliton breathers) in a model based on the nonlinear Schrödinger equation (NLSE), which includes a small quintic term, and weak resonant nonlinearity management (NLM), i.e., time-periodic modulation of the cubic coefficient, at the frequency close to that of shape oscillations of the 2-soliton. The model applies to the light propagation in media with cubic-quintic optical nonlinearities and periodic alternation of linear loss and gain, and to BEC, with the self-focusing quintic term accounting for the weak deviation of the dynamics from one-dimensionality, while the NLM can be induced by means of the Feshbach resonance. We propose an explanation to the effect of the resonant splitting of the 2-soliton under the action of the NLM. Then, using systematic simulations and an analytical approach, we conclude that the weak quintic nonlinearity with the self-focusing sign stabilizes the 2-soliton, while the selfdefocusing quintic nonlinearity accelerates its splitting. It is also shown that the quintic term with the self-defocusing/focusing sign makes the resonant response of the 2-soliton to the NLM essentially broader, in terms of the frequency.

2012

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrodinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons. First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanica...