Two-parameter determinant representation of seventh order rogue wave solutions of the NLS equation

2013

We present a new representation of solutions of focusing nonlinear Schrodinger equation (NLS) equation as a quotient of two determinants. We construct families of quasi-rational solutions of the NLS equation depending on two parameters, a and b. We construct, for the first time, analytical expressions of Peregrine breather of order 7 and multi-rogue waves by deformation of parameters. These expressions make possible to understand the behavior of the solutions. In the case of the Peregrine breather of order 7, it is shown for great values of parameters a or b the appearance of the Peregrine breather of order 5. 35Q55; 37K10

Rational solutions to the KPI equation from particular polynomials

2022

Abstract We construct solutions to the Kadomtsev–Petviashvili equation (KPI) from particular polynomials. We obtain rational solutions written as a second spatial derivative of a logarithm of a determinant of order n . We obtain with this method an infinite hierarchy of rational solutions to the KPI equation. We give explicitly the expressions of these solutions for the first five orders.

Hierarchy of solutions to the NLS equation and multi-rogue waves.

2014

The solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) are given in terms of determinants. The orders of these determinants are arbitrarily equal to 2N for any nonnegative integer $N$ and generate a hierarchy of solutions which can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N(N+1) in x and t. These solutions depend on 2N-2 parameters and can be seen as deformations with 2N-2 parameters of the Peregrine breather P_{N} : when all these parameters are equal to 0, we recover the P_{N} breather whose the maximum of the module is equal to 2N+1. Several conjectures about the structure of the solutions are given.

Tenth Peregrine breather solution to the NLS equation

2015

We go on in this paper, in the study of the solutions of the focusing NLS equation. With a new representation given in a preceding paper, a very compact formulation without limit as a quotient of two determinants, we construct the Peregrine breather of order N=10. The explicit analytical expression of the Akhmediev's solution is completely given.

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

2015

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,…

A New Family of Deformations of Darboux-Pöschl-Teller Potentials

2004

The aim of this Letter is to present a new family of integrable functional-difference deformations of the Schrodinger equation with Darboux–Poschl–Teller potentials. The related potentials are labeled by two integers m and n, and also depend on a deformation parameter h. When h→ 0 the classical Darboux–Poschl–Teller model is recovered.

Wronskian representation of solutions of NLS equation, and seventh order rogue wave.

2012

This work is a continuation of a recent paper in which we have constructed a multi-parametric family of the nonlinear Schrodinger equation in terms of wronskians. When we perform a special passage to the limit, we get a family of quasi-rational solutions expressed as a ratio of two determinants. We have already construct Peregrine breathers of orders N=4, 5, 6 in preceding works; we give here the Peregrine breather of order seven.

Rational solutions to the mKdV equation associated to particular polynomials

2021

International audience; Rational solutions to the modified Korteweg-de Vries (mKdV) equation are given in terms of a quotient of determinants involving certain particular polynomials. This gives a very efficient method to construct solutions. We construct very easily explicit expressions of these rational solutions for the first orders n = 1 until 10.

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

2009

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.

Rational solutions to the KPI equation and multi rogue waves

2016

Abstract We construct here rational solutions to the Kadomtsev–Petviashvili equation (KPI) as a quotient of two polynomials in x , y and t depending on several real parameters. This method provides an infinite hierarchy of rational solutions written in terms of polynomials of degrees 2 N ( N + 1 ) in x , y and t depending on 2 N − 2 real parameters for each positive integer N . We give explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the ( x , y ) plane for different values of time t and parameters.

Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

2013

For the famous Darboux-Pöschl-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.

Higher order Peregrine breathers solutions to the NLS equation

2015

The solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N (N + 1) in x and t. These solutions depend on 2N − 2 parameters : when all these parameters are equal to 0, we obtain the famous Peregrine breathers which we call PN breathers. Between all quasi-rational solutions of the rank N fixed by the condition that its absolute value tends to 1 at infinity and its highest maximum is located at the point (x = 0, t = 0), the PN breather is distinguished by the fact that PN (0, 0) = 2N + 1. We construct Peregrine breathers of the rank N explicitly for N ≤ 11. We give …

Degenerate Riemann theta functions, Fredholm and wronskian representations of the solutions to the KdV equation and the degenerate rational case

2021

International audience; We degenerate the finite gap solutions of the KdV equation from the general formulation given in terms of abelian functions when the gaps tend to points, to get solutions to the KdV equation given in terms of Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, Fredholm determinants and wronskians. This gives the bridge between the algebro-geometric approach and the Darboux dressing method.We construct also multi-parametric degenerate rational solutions of this equation.

The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation.

2013

We construct here explicitly new deformations of the Peregrine breather of order 5 with 8 real parameters. This gives new families of quasi-rational solutions of the NLS equation and thus one can describe in a more precise way the phenomena of appearance of multi rogue waves. With this method, we construct new patterns of different types of rogue waves. We get at the same time, the triangular configurations as well as rings isolated. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings.

Deformations of third-order Peregrine breather solutions of the nonlinear Schrödinger equation with four parameters

2013

We present a new representation of solutions of the one-dimensional nonlinear focusing Schr\"odinger equation (NLS) as a quotient of two determinants. This formulation gives in the case of the order 3, new solutions with four parameters. This gives a very efficient procedure to construct families of quasirational solutions of the NLS equation and to describe the apparition of multirogue waves. With this method, we construct analytical expressions of four-parameters solutions; when all these parameters are equal to 0, we recover the Peregrine breather of order 3. It makes possible with this four-parameters representation, to generate all the types of patterns for the solutions, like the tria…

The Peregrine breather of order nine and its deformations with sixteen parameters solutions to the NLS equation

2015

Abstract We construct new deformations of the Peregrine breather ( P 9 ) of order 9 with 16 real parameters. With this method, we obtain explicitly new families of quasi-rational solutions to the NLS equation in terms of a product of an exponential depending on t by a ratio of two polynomials of degree 90 in x and t; when all the parameters are equal to 0, we recover the classical P 9 breather. We construct new patterns of different types of rogue waves as triangular configurations of 45 peaks as well as rings and concentric rings.

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

2015

We present here new solutions of the focusing one-dimensional nonlinear Schrödinger (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.

The mKdV equation and multi-parameters rational solutions

2021

Abstract N -order solutions to the modified Korteweg–de Vries (mKdV) equation are given in terms of a quotient of two wronskians of order N depending on 2 N real parameters. When one of these parameters goes to 0, we succeed to get for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2 N real parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 6 .

Rational solutions to the Johnson equation of order N depending on 2N − 2 parameters

2019

We construct rational solutions of order N depending on 2N − 2 parameters. They can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters. We explicitly construct the expressions of the rational solutions of order 4 depending on 6 real parameters and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, b1, b2, b3.

Solutions to the NLS equation : differential relations and their different representations

2020

Solutions to the focusing nonlinear Schrödinger equation (NLS) of order N depending on 2N − 2 real parameters in terms of wronskians and Fredholm determinants are given. These solutions give families of quasirational solutions to the NLS equation denoted by vN and have been explicitly constructed until order N = 13. These solutions appear as deformations of the Peregrine breather PN as they can be obtained when all parameters are equal to 0. These quasi rational solutions can be expressed as a quotient of two polynomials of degree N (N + 1) in the variables x and t and the maximum of the modulus of the Peregrine breather of order N is equal to 2N + 1. Here we give some relations between sol…

Quasi-rational solutions of the NLS equation and rogue waves

2010

We degenerate solutions of the NLS equation from the general formulation in terms of theta functions to get quasi-rational solutions of NLS equations. For this we establish a link between Fredholm determinants and Wronskians. We give solutions of the NLS equation as a quotient of two wronskian determinants. In the limit when some parameter goes to $0$, we recover Akhmediev's solutions given recently It gives a new approach to get the well known rogue waves.

Rational Solutions to the KdV Equation in Terms of Particular Polynomials

2022

Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order n for any positive integer n, and we call these solutions, solutions of the order n. Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order 10.

Multi-parameters rational solutions to the mKdV equation

2021

N-order solutions to the modified Korteweg-de Vries (mKdV) equation are given in terms of a quotient of two wronskians of order N depending on 2N real parameters. When one of these parameters goes to 0, we succeed to get for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N real parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 6.

Solutions of the LPD equation and multi-parametric rogue waves

2022

Quasi-rational solutions to the Lakshmanan Porsezian Daniel equation are presented. We construct explicit expressions of these solutions for the first orders depending on real parameters. We study the patterns of these configurations in the (x, t) plane in function of the different parameters. We observe in the case of order 2, three rogue waves which move according to the two parameters. In the case of order 3, six rogue waves are observed with specific configurations moving according to the four parameters.

Other patterns for the first and second order rational solutions to the KPI equation

2022

We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree 2N (N + 1) − 2 in x, y and t by a polynomial of degree 2N (N + 1) in x, y and t, depending on 2N − 2 real parameters for each positive integer N. We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x, y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of…

The defocusing NLS equation : quasi-rational and rational solutions

2022

Quasi-rational solutions to the defocusing nonlinear Schrödinger equation (dNLS) in terms of wronskians and Fredholm determinants of order 2N depending on 2N − 2 real parameters are given. We get families of quasi-rational solutions to the dNLS equation expressed as a quotient of two polynomials of degree N (N + 1) in the variables x and t. We present also rational solutions as a quotient of determinants involving certain particular polynomials.

Fredholm representations of solutions to the KPI equation, their wronkian versions and rogue waves

2016

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions called solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N (N + 1) in x, y and t depending on 2N − 2 parameters. So we get with this method an infinite hierarchy of solutions to the KPI equation.

Deformations of third order Peregrine breather solutions of the NLS equation with four parameters

2013

In this paper, we give new solutions of the focusing NLS equation as a quotient of two determinants. This formulation gives in the case of the order 3, new deformations of the Peregrine breather with four parameters. This gives a very efficient procedure to construct families of quasi-rational solutions of the NLS equation and to describe the apparition of multi rogue waves. With this method, we construct the analytical expressions of deformations of the Peregrine breather of order N=3 depending on $4$ real parameters and plot different types of rogue waves.

Multi-parametric families solutions to the Burgers equation

2021

We construct 2N real parameter solutions to the Burgers' equation in terms of determinant of order N and we call these solutions, N order solutions. We deduce general expressions of these solutions in terms of exponentials and study the patterns of these solutions in functions of the parameters for N = 1 until N = 4.

8-parameter solutions of fifth order to the Johnson equation

2019

We give different representations of the solutions of the Johnson equation with parameters. First, an expression in terms of Fredholm determinants is given; we give also a representation of the solutions written as a quotient of wronskians of order 2N. These solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polyno-mials of degree 2N (N +1) in x, t and 4N (N +1) in y depending on 2N −2 parameters. Here, we explicitly construct the expressions of the rational solutions of order 5 depending on 8 real parameters and we study the patterns of their modulus in the plane (x, y) and their …

Fredholm and wronskian representations of solutions to the Johnson equation and the third order case

2019

We construct solutions to the Johnson equation (J) by means of Fred-holm determinants first, then by means of wronskians of order 2N giving solutions of order N depending on 2N − 1 parameters. We obtain N order rational solutions which can be written as a quotient of two polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x, y) plane in function of time t and parameters a1, a2, b1, b2.

Le Graphe Génératif Gaussien

2009

International audience; Un nuage de points est plus qu'un ensemble de points isolés. La distribution des points peut être gouvernée par une structure topologique cachée, et du point de vue de la fouille de données, modéliser et extraire cette structure est au moins aussi important que d'estimer la seule densité de probabilité du nuage. Dans cet article, nous proposons un modèle génératif basé sur le graphe de Delaunay d'un ensemble de prototypes représentant le nuage de points, et supposant un bruit gaussien. Nous dérivons un algorithme pour la maximisation de la vraisemblance des paramètres, et nous utilisons le critère BIC pour sélectionner la complexité du modèle. Ce travail a pour objec…

Solutions to the Gardner equation with multi-parameters and the rational case

2022

We construct solutions to the Gardner equation in terms of trigonometric and hyperbolic functions, depending on several real parameters. Using a passage to the limit when one of these parameters goes to 0, we get, for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 3. We easily deduce solutions to the mKdV equation in terms of wronskians as well as rational solutions depending on 2N real parameters.

N order solutions with multi-parameters to the Boussinesq and KP equations and the degenerate rational case

2021

From elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these parameters goes to 0, we get rational solutions as a quotient of a polynomial of degree N (N + 1) − 2 in x and t, by a polynomial of degree N (N + 1) in x and t for each positive integer N depending on 3N parameters. We restrict ourself to give the explicit expressions of these rational solutions for N = 1 until N = 3 to shortened the paper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from 1 to 3.

Solutions to the Gardner equation with multiparameters and the rational case

2022

We construct solutions to the Gardner equation in terms of trigonometric and hyperbolic functions, depending on several real parameters. Using a passage to the limit when one of these parameters goes to 0, we get, for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 3. We easily deduce solutions to the mKdV equation in terms of wronskians as well as rational solutions depending on 2N real parameters.

Quasi-rational solutions of the Hirota equation depending on multi-parameters and rogue waves

2023

Quasi-rational solutions to the Hirota equation are given. We construct explicit expressions of these solutions for the first orders. As a byproduct, we get quasi-rational solutions to the focusing NLS equation and also rational solutions to the mKdV equation. We study the patterns of these configurations in the (x, t) plane.

Families of deformations of the thirteen peregrine breather solutions to the NLS equation depending on twenty four parameters

2017

International audience; We go on with the study of the solutions to the focusing one dimensional nonlinear Schrodinger equation (NLS). We construct here the thirteen's Peregrine breather (P13 breather) with its twenty four real parameters, creating deformation solutions to the NLS equation. New families of quasirational solutions to the NLS equation in terms of explicit ratios of polynomials of degree 182 in x and t multiplied by an exponential depending on t are obtained. We present characteristic patterns of the modulus of these solutions in the (x; t) plane, in function of the different parameters.

From particular polynomials to rational solutions to the KPI equation

2022

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) from particular polynomials. We obtain rational solutions written as a second derivative with respect to the variable x of a logarithm of a determinant of order n. So we get with this method an infinite hierarchy of rational solutions to the KPI equation. We give explicitly the expressions of these solutions for the first five orders.

Rational solutions to the KPI equation of order 7 depending on 12 parameters

2018

We construct in this paper, rational solutions as a quotient of two determinants of order 2N = 14 and we obtain what we call solutions of order N = 7 to the Kadomtsev-Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 112 in x, y and t depending on 12 parameters. The maximum of modulus of these solutions at order 7 is equal to 2(2N + 1) 2 = 450. We make the study of the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6. When all these parameters grow, triangle and ring structures are obtained.

An extended Darboux transformation to get families of solutions to the KPI equation

2023

By means of a Darboux transform with particular generating function solutions to the Kadomtsev-Petviashvili equation (KPI) are constructed. We give a method that provides different types of solutions in terms of particular determinants of order N. For any order, these solutions depend of the degree of summation and the degree of derivation of the generating functions. We study the patterns of their modulus in the plane (x, y) and their evolution according time and parameters.

2N+1 highest amplitude of the modulus of the N-th order AP breather and other 2N-2 parameters solutions to the NLS equation

2015

We construct here new deformations of the AP breather (Akhmediev-Peregrine breather) of order N (or AP N breather) with 2N −2 real parameters. Other families of quasi-rational solutions of the NLS equation are obtained. We evaluate the highest amplitude of the modulus of AP breather of order N ; we give the proof that the highest amplitude of the AP N breather is equal to 2N + 1. We get new formulas for the solutions of the NLS equation, 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 get the triangular configurations as well as isolated rings at the same time. Moreover, the appea…

Tenth order solutions to the NLS equation with eighteen parameters

2015

We present here new solutions of the focusing one dimensional non linear Schrödinger 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 as found for the lower orders.

Higher order Peregrine breathers and multi-rogue waves solutions of the NLS equation

2012

This work is a continuation of a recent paper in which we have constructed a multi-parametric family of solutions of the focusing NLS equation given 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 get a family of quasi-rational solutions. Here we construct Peregrine breathers of orders N=4, 5, 6 and the multi-rogue waves corresponding in the frame of the NLS model first explained by Matveev et al. in 2010. In the cases N=4, 5, 6 we get convenient formulas to study the deformation of higher Peregrine breather of order 4, 5 and 6 to the multi-rogue waves via…

Families of solutions to the KPI equation and the structure of their rational representations of order N

2018

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions called solutions of order N depend on 2N − 1 parameters. They can also be written as a quotient of two polynomials of degree 2N (N + 1) in x, y and t depending on 2N − 2 parameters. The maximum of the modulus of these solutions at order N is equal to 2(2N + 1) 2. We explicitly construct the expressions until the order 6 and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters.

Families of solutions of order nine to the NLS equation with sixteen parameters

2015

We construct new deformations of the Peregrine breather (P9) of order 9 with 16 real parameters. With this method, we obtain explicitly new families of quasi-rational solutions to the NLS equation in terms of a product of an exponential depending on t by a ratio of two polynomials of degree 90 in x and t; when all the parameters are equal to 0, we recover the classical P9 breather. We construct new patterns of different types of rogue waves as triangular configurations of 45 peaks as well as rings and concentric rings.

Rational solutions to the KdV equation depending on multi-parameters

2021

We construct multi-parametric rational solutions to the KdV equation. For this, we use solutions in terms of exponentials depending on several parameters and take a limit when one of these parameters goes to 0. Here we present degenerate rational solutions and give a result without the presence of a limit as a quotient of polynomials depending on 3N parameters. We give the explicit expressions of some of these rational solutions.

Eighth Peregrine breather solution of the NLS equation and multi-rogue waves

2012

This is a continuation of a paper in which we present a new representation of solutions of the focusing NLS equation as a quotient of two determinants. This work was based on a recent paper in which we had constructed a multi-parametric family of this equation in terms of wronskians. \\ Here we give a more compact formulation without limit. With this method, we construct Peregrine breather of order N=8 and multi-rogue waves associated by deformation of parameters.

From particular polynomials to rational solutions to the PII equation

2022

The Painlevé equations were derived by Painlevé and Gambier in the years 1895 − 1910. Given a rational function R in w, w ′ and analytic in z, they searched what were the second order ordinary differential equations of the form w ′′ = R(z, w, w ′) with the properties that the singularities other than poles of any solution or this equation depend on the equation only and not of the constants of integration. They proved that there are fifty equations of this type, and the Painlevé II is one of these. Here, we construct solutions to the Painlevé II equation (PII) from particular polynomials. We obtain rational solutions written as a derivative with respect to the variable x of a logarithm of a…

Families of solutions to the CKP equation with multi-parameters

2020

We construct solutions to the CKP (cylindrical Kadomtsev-Petviashvili)) equation in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions are called solutions of order N ; they depend on 2N − 1 parameters. They can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters. We explicitly construct the expressions up to order 5 and we study the patterns of their modulus in plane (x, y) and their evolution according to time and parameters.

Ten-parameters deformations of the sixth order Peregrine breather solutions of the NLS equation.

2013

In this paper, we construct new deformations of the Peregrine breather of order 6 with 10 real parameters. We obtain new families of quasi-rational solutions of the NLS equation. With this method, we construct new patterns of different types of rogue waves. We get as already found for the lower order, the triangular configurations and rings isolated. Moreover, one sees for certain values of the parameters the appearance of new configurations of concentric rings.

Patterns of deformations of P 3 and P 4 breathers solutions to the NLS equation

2016

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the (x, t) plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending on t. This formulation gives in the case of the order 3 and 4, solutions with respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others.

Multi-lump solutions to the KPI equation with a zero degree of derivation

2022

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) by using an extended Darboux transform. From elementary functions we give a method that provides different types of solutions in terms of wronskians of order N. For a given order, these solutions depend on the degree of summation and the degree of derivation of the generating functions.In this study, we restrict ourselves to the case where the degree of derivation is equal to 0. In this case, we obtain multi-lump solutions and we study the patterns of their modulus in the plane (x,y) and their evolution according time and parameters.

Riemann theta functions, Fredholm and wronskian representations of the solutions to the KdV equation

2021

We degenerate the finite gap solutions of the KdV equation from the general formulation given in terms of abelian functions when the gaps tends to points, to get solutions to the KdV equation given in terms of Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, Fredholm determinants and wronskians. This gives the bridge between the algebro-geometric approach and the Darboux dressing method.

Particular polynomials generating rational solutions to the KdV equation

2022

We construct here rational solutions to the KdV equation by means of particular polynomials. We get solutions in terms of determinants of order n for any positive integer n and we call these solutions, solutions of order n. So we obtain a very efficient method to get rational solutions to the KdV equation and we can construct very easily explicit solutions. In the following, we present some solutions until order 10.

Zero degree of derivation for multi-lump solutions to the KPI equation

2023

From first to fourth order rational solutions to the Boussinesq equation

2020

Rational solutions to the Boussinesq equation are constructed as a quotient of two polynomials in x and t. For each positive integer N , the numerator is a polynomial of degree N (N + 1) − 2 in x and t, while the denominator is a polynomial of degree N (N + 1) in x and t. So we obtain a hierarchy of rational solutions depending on an integer N called the order of the solution. We construct explicit expressions of these rational solutions for N = 1 to 4.

From particular polynomials to rational solutions to the mKdV equation

2022

Rational solutions to the modified Korteweg-de Vries (mKdV) equation are given in terms of a quotient of determinants involving certain particular polynomials. This gives a very efficient method to construct solutions. We construct very easily explicit expressions of these rational solutions for the first orders n = 1 until 10.