Stabilization of solitons of nonlinear Schrödinger equations
Estado Finalizada
Trabajo fin de master y Tesis doctoral
Director UCLM
Montesinos Matilla, Gaspar

In Chapter 1 I have presented a brief overview of the nonlinear Schr ?odinger equation (NLS), mainly in the case of power-type nonlinearity. I have introduced historically the standing waves or solitary waves as well as the solitonic solutions. The structural prop- erties of the NLS equation have been shown. The Lagrangian one is the starting point of an usual approach to study properties of solutions to the NLS equation (the method of collective coordinates or the variational approach), which has been used through the work. I have also discussed the phenomenon of blowup or collapse which is related to the global existence of the solutions. Finally, a rigorous definition and some properties of standing waves have been reviewed.

In Chapter 2 I have shown the method of moments in the framework of general NLS equations. I have derived the equations of the method and found all the nonlinearities for which the method gives exact results. The quadratic phase approximation method, based on the method of moments, is also discussed in the chapter and applied to different problems, such as the writing of simple equations describing the stabilization of solitonic structures by control of the nonlinear term and the dynamics of structures in cubic-quintic media.

In Chapter 3 I have studied the phenomenon of stabilization of solitons of the cubic NLS equation obtained by a periodic control of the nonlinear term. I have applied the moment equations derived in the previous chapter to two- and three-dimensional systems and obtained, on the basis of their rigorous analysis, precise conditions for the stabilization of two dimensional systems.

Taking as initial data Townes solitons or Gaussians for simulations based on the NLS equation, I have shown that the former is the structure which is stabilized and that other initial data, which can be stabilized, eject a fraction of the wave packet adapting the shape of the Townes soliton.

I have also analyzed the three-dimensional situation. Here I have made an extensive search of stable regions according to the moment equations improving and extending the analysis of Ref. [111]. I have justified why moment-type equations cannot be used to predict the dynamics of the system. Only limited time stabilization is possible when the parameters are fine-tuned to very precise values. Finally, when a strong trapping along one specific direction is kept, the system becomes effectively two-dimensional and it can be described again by variational methods, whose predictions agree well with the full numerical simulations of the problem. In the latter case, three-dimensional confinement is possible although there are only two spatial directions along which the solution is trapped, by the nonlinear forces plus the stabilization mechanisms, while the other direction is trapped by harmonic forces provided by the potential.

The existence of stabilized solitons is a remarkable phenomenon which opens new fields for applications, in fact these are the first stable structures obtained in the framework of the multidimensional cubic nonlinear Schr ?odinger equation.

As reported in the chapter and Ref. [111] stabilization of three-dimensional structures seem not to be possible. A recent work (see Ref. [179]) points out the success in the stabilization of such structures. Nevertheless, in Ref. [180] it is concluded that the stabi- lization in that three-dimensional scenario may not be due to the oscillating nonlinearity, but due to a delicate balance between the kinetic term and the attractive nonlinearity.

In Ref. [181] the global well-posedness of solutions of the averaged NLS equation that describe strong nonlinearity modulation of the NLS equation is studied. There, it is shown that the blowup of multidimensional solutions is arrested within the averaged NLS equation. It is an open problem to study well-posedness of the full NLS equation, depending on parameters of the nonlinearity modulation and profile of initial data.

In Ref. [180] a model including a dissipation term in the equation allows stabilization of three-dimensional structures. However, this phenomenological loss term is no realistic and has no practical relevance. Therefore, the possibility of achieving stabilization with realistic dissipation is an open question for future work.

In Chapter 4 I have presented a new type of vector solitons, the stabilized vector solitons (SVS). I have derived an effective-particle model based on Lagrangian methods to study the interactions of the stabilized Townes solitons (STS) shown in Chapter 3. I have studied their dynamics by direct numerical simulations of the model equations and found that, in the regime of fast collisions, the solitons emerge with slight modifications. On the other hand I have shown that the SVSs arise when slow collisions of STSs take place and a quasi-bound state can be formed at very low velocities. Moreover the system presents collapsing and expanding orbits depending on the initial configuration as well as other phenomena, as deflection of one wave packet due to the attraction and split in several parts in the corresponding n-body interactions.

I have also presented a theoretical explanation of the wave packet splitting based on effective Lagrangian methods, and corroborated that effective-particle models allow to obtain conclusions for this system only in very limited situations.

It is important to point out that some approximate general methods have been devel- oped in previous works (see Refs. [182, 183]) to find an effective potential of interaction between far separated solitons, both in scalar and vector systems, when the nonlinear coefficient is constant. In our case, however, the fact that the nonlinear coefficient is a function of time and that the solitons are interacting, leading to the formation of complex structures, makes a similar approach not valid for our work. Developing some kind of approach to find an effective potential of interaction would be an interesting extension of our work.

In Chapter 5 I have studied the possibility of stabilizing solutions with angular momen- tum (vortices) propagating in Kerr media against filamentation and collapse. Following the ideas of the previous chapters the procedure used consists in taking an appropriate layered medium, with alternating focusing and defocusing nonlinearities, so that one can retard the filamentation of the beam. Nevertheless, the addition of a slight noise makes the beam break into filaments very early, so the stabilization mechanism is not valid in practical situations. One form of avoiding this and obtaining long propagation distances is to use an incoherent guiding beam previously stabilized, which acts as a trapping po- tential for the vortex. I have also discussed the practical implementation of this stabilizing mechanism in nonlinear optics and Bose-Einstein condensates.

The possibility of stabilizing different solutions from the Townes soliton and vortex- type ones is an open question which has been started in Ref. [184] for one-node positive solutions of the NLS equation. Nevertheless, more work is needed to study other solutions such as multi-node positive solutions or more complex ones as those of Refs. [185, 186].

In Chapter 6 I have described two-dimensional spatio-temporal solitons which are sta- bilized against collapse by means of dispersion management. I have studied their stability and properties both analytically and numerically. I have shown that the adequate choice of the modulation parameters will optimize the stabilization of the pulse. I have also checked the robustness of the method by strong changes in the modulation function and the addi- tion of noise. The results are of relevance in the field of high power pulse propagation in nonlinear optical materials.

An important extension of this work is the stabilization by means of dispersion man- agement in a model including loss and gain terms. These stabilized structures would be the analogous two-dimensional of the guiding-center solitons [187, 188].

A remaining open question concerns the stability of higher-dimensional beams. In fact, at the present time, it is not clear the mechanism of stabilization of fully three-dimensional solitons which have been predicted to collapse in the case of modulation of the nonlinearity [111]. Thus, the results on dispersion management would be an important step forward in the stabilization of light bullets, in combination with modulation of the nonlinearity.