Research
KP equations

A central theme in my research is the study of nonlinear partial differential equations by means of the so-called Inverse Scattering (or Spectral) transform method. 

Part of my research activity, in collaboration with proff. M. Boiti, F. Pempinelli (Lecce) and A.K. Pogrebkov (Steklov Institute, Moscow), deals with the development of a new mathematical technique which allows to extend the spectral transform for KP equations (and possibly of other nonlinear 2+1 dimensional equations relevant for physical applications) to the case of potentials that are not decaying along a finite number of directions in the plane (ray type potentials). Such extension of the class of the admissible potentials is of great mathematical relevance since it would provide the possibility to include solutions with one-dimensional asymptotic behavior. In developing the inverse scattering theory in two dimensions one has to face new problems which are intrinsically related to the multidimensional character of the theory. Moreover, the spectral theory for non-decaying potentials is essentially more involved than for the standard case of rapidly decaying potentials. Recently it has been discovered that it is convenient to study the spectral theory of the principal Lax operator defining the scattering problem in the space of the Fourier transform of solutions, since the spectral data have a very natural expression in terms of variables in this space. Starting from this basic idea it was built the so-called extended resolvent approach. It is a generalization to complex variables of the usual resolvent and one of its attracting features is that all the quantities which are relevant for IST technique (Green’s function, Jost solutions, spectral data etc.) are obtained quite straightforwardly from the resolvent through proper regularizations and reductions. When one admits non-decaying potentials in order to include also wave solitons, the theory is complicated by the fact that the potential has to be regarded no longer as a function in the Schwartz space, but as a distribution in the dual space. As a consequence it is necessary to introduce a regularization procedure which allows to define products of distributions in two variables that naturally appear in the theory. In spite of these difficulties, the extended resolvent appears to be the effective foundation for the proper generalization of the inverse scattering method. Indeed, in this framework it was possible to obtain, under some partially implicit assumptions, some general results for potentials of ray type.

NLS systems

For the last 5 years I have been working in collaboration with prof. Mark J. Ablowitz, at the Department of Applied Mathematics of CU Boulder. Our common research activity mainly focuses on the IST for continuous and discrete Nonlinear Schrodinger systems (NLS), both scalar and vector.

NLS equations have become amongst the most important nonlinear systems studied in mathematics and physics. Such systems have been derived in such diverse fields as deep water waves, plasma physics, nonlinear fiber optics, magnetic spin waves, etc. In fact, it turns out that almost any dispersive, energy preserving system gives rise, in an appropriate limit, to the scalar NLS equation. Its vector generalization (VNLS) arises, physically, under conditions similar to those described by the NLS when there are multiple wave-trains moving with nearly the same group velocities. Also, VNLS models physical systems in which the field has more than one component; for example, in optical fibers and waveguides, the propagating electric field has two components transverse to the direction of propagation. Both the NLS and VNLS equations admit integrable discretizations that besides being used as the basis for constructing numerical schemes for their continuous counterparts, also have physical applications as discrete systems. Moreover, both the scalar and vector NLS equations and some of their discretizations attain broad significance because they are integrable via the inverse scattering transform (IST) and admit soliton solutions.
In spite of the fact that NLS systems have been the subject of investigation by reaserchers for over 30 years, many important problems remain to be solved. During the last years we have devoted our efforts to studying some of them.
In particular, we investigated soliton interactions. It was well-known that interacting scalar solitons affect each other only by a phase shift, that depends only on the soliton powers and velocities, both of which are conserved quantities. Thus, when two soliton collisions occur sequentially, the outcome of the first collision does not affect the second collision, except for a uniform phase shift. On the other hand, the collision of vector solitons is much richer and more complex, due to the fact that vector and matrix solitons have internal degrees of freedom. In fact, while the dynamics of a single soliton is essentially governed by the underlying scalar equation, the vector nature of the soliton manifests itself in the interaction, which can be highly nontrivial: even though the collision is elastic, in the sense that the total energy of each soliton is conserved, there can be a significant redistribution of energy among the components, i.e. a polarization shift. We gave explicit formulas for the polarization shift induced by the soliton collision, both for the continuous VNLS equation and, for the first time, for its integrable discretization. Even for the classical VNLS, in spite of the fact that the system had been known for over 30 years, the nature of multisoliton collisions was not clarified. For the first time, revisiting and casting more light on Manakov's well-known results, we were able to prove rigorously that for the vector systems, both continuous and discrete, the multisoliton interaction process is pair-wise and the net result of the interaction is independent of the order in which such collisions occur.
We analyzed the various features underlying collisions of generic N-component VNLS solitons, including changes in amplitudes, phases and relative separation distances. Then we established a relationship between order independence of the pair-wise multisoliton interaction processes and the fact that the map determining the interaction of two solitons satisfies the Yang-Baxter relation,  where the same problem was addressed for the matrix KdV equation. In particular, we showed that the (matrix) transmission coefficients of the VNLS equation satisfy the matrix re-factorization problem which leads to the Yang-Baxter relation.
The interaction of vector solitons also sets forth the experimental foundations for computation with solitons. In the energy re-distributions in the two component case was expressed as fractional linear transformations (FLT's), showing that the parameters controlling the energy switching between components exhibit nontrivial information transformations. As a result, sequences of solitons operating on other sequences of solitons that effect logic operations, including controlled gates, can be implemented. Using this background, we analyzed multisoliton interactions in the general N-component case and expressed them explicitly via generalized FLT's, where a soliton is described in terms of N-1 complex parameters. Finally, we considered the basic (three collision) gate introduced by Steiglitz et al, where it was shown that in order to implement it, one has to be able to adjust the values of the initial polarizations of two of the solitons participating in the gate according to a given system of equations. Only numerical solutions where given and the issue of when they do and do not have solutions was left open. We proved that there are parameter regimes for the soliton amplitudes and velocities for which a unique solution for the system exists.
We showed that the integrable discretization of the VNLS (IDVNLS) equation admits soliton solutions. In fact,there are both "fundamental'' soliton solutions, that are the counterpart of the vector solitons of VNLS, and "composite'' soliton solutions that have no counterpart in the continuous limit. The composite soliton itself consists of a localized traveling envelope with both temporal and spatial oscillations, as well as a complex spatial modulation. In order to shed light on the structure of composite solitons, we considered in more detail two special cases of the composite soliton, which we referred to as the "orthogonal'' and the "parallel'' case. An orthogonal composite soliton is the (weighted) sum of two fundamental-soliton-like solutions. The two constituent sech-envelopes have mutually orthogonal polarizations. Moreover, one of the envelopes has oscillation on the grid scale, hence, there is no continuum limit of this solution. On the other hand, the parallel composite soliton is a scalar function multiplied by a polarization vector. In fact, the shape of the overall envelope is a superposition of two modulated sech-envelopes.
We studied the interaction of fundamental solitons and showed that the order independence of the multisoliton collisions is a reflection of the fact that, in analogy with their continuous counterparts, discrete vector soliton interactions can be recast as Yang-Baxter maps. Moreover, we showed that, like in the continuous case, the polarization state of a discrete vector soliton can be described by a single complex scalar and the polarization shifts induced by soliton collisions is conveniently represented by means of fractional linear transformations (FLTs) on these scalars. The remarkable consequence is that interacting discrete vector solitons exhibit nontrivial information transfer and the value of a logical ``bit'' can be encoded in the scalar polarization state of a vector soliton. Given such an encoding, one can identify vector-soliton interactions with logical operations. Then, using the model proposed by Steiglitz et al for the Manakov spatial solitons, we were able to show that the spatially-discrete vector solitons of two-component IDVNLS are also computationally universal in the sense of Turing equivalence.

Recently, we began to investigate the IST for NLS systems with nonvanishing boundary conditions at space infinities. While the IST for the scalar NLS, both with vanishing and nonvanishing boundaries, was developed many years ago, the basic formulation of direct and inverse scattering for the vector NLS equation in the normal dispersion regime (defocusing VNLS) has still not been given in the case of nonvanishing boundary conditions as | x| tends to infinity. The defocusing VNLS equation admits solitons with a nontrivial background intensity in either one or both components, so-called dark-bright or dark-dark solitons, which makes formulating the IST for nondecaying potentials particularly interesting.


 

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