Pierre Gaillard

FAMILIES OF QUASI-RATIONAL SOLUTIONS OF THE NLS EQUATION AND MULTI-ROGUE WAVES

We construct a multi-parametric family of the solutions of the focusing nonlinear Schr?dinger equation (NLS) from the known results describing the multi-phase almost-periodic elementary solutions given in terms of Riemann theta functions. We give a new representation of their solutions in terms of Wronskians determinants of order 2N composed of elementary trigonometric functions. When we perform a special passage to the limit when all the periods tend to infinity, we obtain a family of quasi-rational solutions. This leads to efficient representations for the Peregrine breathers of orders N = 1,?2,?3 first constructed by Akhmediev and his co-workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue-wave formation in the frame of the NLS model first explained by V B Matveev in 2010. Our formulation allows us to isolate easily the second- or third-order Peregrine breathers from ?generic? solutions and also to compute the Peregrine breathers of orders 2 and 3 easily with respect to other approaches. In the cases N = 2,?3, we obtain the comfortable formulas to study the deformation of a higher Peregrine breather of order 2 to the three rogue-wave or order 3 to the six rogue-wave solutions via the variation of the free parameters of our construction.

Wronskian and Casorati determinant representations for Darboux–Pöschl–Teller potentials and their difference extensions

We consider some special reductions of generic Darboux?Crum dressing formulae and of their difference versions. As a matter of fact, we obtain some new formulae for Darboux?P?schl?Teller (DPT) potentials by means of Wronskian determinants. For their difference deformations (called DDPT-I and DDPT-II potentials) and the related eigenfunctions, we obtain new formulae described by the ratios of Casorati determinants given by the functional difference generalization of the Darboux?Crum dressing formula.

Other 2N − 2 parameters solutions of the NLS equation and 2N + 1 highest amplitude of the modulus of the N th order AP breather

In this paper, we construct new deformations of the Akhmediev-Peregrine (AP) breather of order N (or APN breather) with real parameters. Other families of quasirational solutions of the nonlinear Schrodinger (NLS) equation are obtained. We evaluate the highest amplitude of the modulus of the AP breather of order N; we give the proof that the highest amplitude of the APN breather is equal to . We get new formulas for the solutions of the NLS equation, which are different from these already given in previous works. New solutions for the order 8 and their deformations according to the parameters are explicitly given. We simultaneously get triangular configurations and isolated rings. Moreover, the appearance for certain values of the parameters and of new configurations of concentric rings are underscored.

Two Parameters Deformations of Ninth Peregrine Breather Solution of the NLS Equation and Multi-Rogue Waves

This paper is a continuation of a recent paper on the solutions of the focusing NLS equation. The representation in terms of a quotient of two determinants gives a very efficient method of determination of famous Peregrine breathers and their deformations. Here we construct Peregrine breathers of order and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.

Twenty Parameters Families of Solutions to the NLS Equation and the Eleventh Peregrine Breather

The Peregrine breather of order eleven (P11 breather) solution to the focusing one-dimensional nonlinear Schrodinger equation (NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given: when all parameters are equal to 0 we recover the famous P11 breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the (x; t) plane, in function of the different parameters.

18 parameter deformations of the Peregrine breather of order 10 solutions of the NLS equation

We present here new solutions of the focusing one-dimensional nonlinear Schrodinger (NLS) equation which appear as deformations of the Peregrine breather of order 10 with 18 real parameters. With this method, we obtain new families of quasi-rational solutions of the NLS equation, and we obtain explicit quotients of polynomial of degree 110 in x and t by a product of an exponential depending on t. We construct new patterns of different types of rogue waves and recover the triangular configurations as well as rings and concentric rings as found for the lower-orders.

Twenty Two Parameter Deformations of the Twelfth Peregrine Breather Solutions to the NLS Equation

The twelfth Peregrine breather (P12 breather) solution to the focusing one dimensional nonlinear Schrodinger equation (NLS) with its twenty two real parameters deformations solutions to the NLS equation are explicitly constructed here. New families of quasi-rational solutions of the NLS equation in terms of explicit quotients of polynomials of degree 156 in x and t by a product of an exponential depending on t are obtained. The patterns of the modulus of these solutions in the

Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

(x; t)

TWO-PARAMETER DETERMINANT REPRESENTATION OF SEVENTH ORDER ROGUE WAVE SOLUTIONS OF THE NLS EQUATION

plane, in function of the different parameters are studied in details.

The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

For the famous Darboux-Poschl-Teller equation, we present new wronskian representation both for the potential and the related eigenfunctions. The simplest application of this new formula is the explicit description of dynamics of the DPT potentials and the action of the KdV hierarchy. The key point of the proof is some evaluation formulas for special wronskian determinant.