Research

We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude as by implementing numerically the corresponding Inverse Scattering Transform (IST). The Riemann--Hilbert problem associated to the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift-Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the (x,t)-plane depending on whether |x/(2t)| is less than or bigger than the nonzero amplitude. 

In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schrödinger equation as the number, N, of solitons grows to infinity. We discover configurations of N-soliton solutions which exhibit the formation (as N tends to infinity) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from 0, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of x and t. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with N, and the asymptotic description only included elliptic waves in the long-time asymptotics. 

The complex coupled short-pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so-called self-symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long-time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well-known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process. 

In this paper, we develop the Riemann–Hilbert approach to the inverse scattering transform (IST) for the complex coupled short-pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real short-pulse equation (SPE) and complex short-pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one-soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather. 

This work deals with a class of square matrix nonlinear Schrödinger (MNLS) systems whose reductions include two equations that model hyperfine spin F=1  spinor Bose–Einstein condensates in the focusing and defocusing dispersion regimes, and two novel (mixed sign) equations that were recently shown to be integrable. Our main goal is to discuss the bright soliton solutions and their interactions for the focusing MNLS and for the two mixed sign systems within the framework of the inverse scattering transform. The nature of the solitons and their interactions depend on whether the associated norming constants (polarization matrices) are rank-one matrices (giving rise to ferromagnetic solitons) or full rank (corresponding to polar solitons). By computing the long-time asymptotics of the 2-soliton solutions, we determine how the polarization matrix of each soliton changes because of the interaction. Explicit formulas for the soliton interactions are given for all possible types of interacting solitons, namely ferromagnetic–ferromagnetic, polar–polar, and polar–ferromagnetic soliton interactions, and for all three inequivalent reductions of the MNLS systems that admit regular bright soliton solutions. We also present bound states, representing 2 solitons travelling with the same velocity, for all three systems. 

We consider two independent Wishart matrices A and B, and we analyze the probability distribution of the largest eigenvalue of the matrix (A+B)^(-1)B, where the number of rows in the underlying data matrix for A increases faster than the number of columns, and the dimensions of B grow proportionally. We show that, under this new scaling, the distribution of the largest eigenvalue converges to the Tracy-Widom law. Our analysis leverages a Riemann-Hilbert formulation for Jacobi polynomials with modified weight function. This result extends the reach of universality of the Tracy-Widom distribution in this novel regime.