Search results for "Rogue waves"
showing 10 items of 35 documents
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.
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.
Tenth Peregrine breather solution of the NLS equation.
2012
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.
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.
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.
The Peregrine soliton in nonlinear fibre optics
2010
International audience; The Peregrine soliton is a localized nonlinear structure predicted to exist over 25 years ago, but not so far experimentally observed in any physical system. It is of fundamental significance because it is localized in both time and space, and because it defines the limit of a wide class of solutions to the nonlinear Schrödinger equation (NLSE). Here, we use an analytic description of NLSE breather propagation to implement experiments in optical fibre generating femtosecond pulses with strong temporal and spatial localization, and near-ideal temporal Peregrine soliton characteristics. In showing that Peregrine soliton characteristics appear with initial conditions th…
Shallow water rogue wavetrains in nonlinear optical fibers
2013
International audience; In addition to deep-water rogue waves which develop from the modulation instability of an optical CW, wave propagation in optical fibers may also produce shallow water rogue waves. These extreme wave events are generated in the modulationally stable normal dispersion regime. A suitable phase or frequency modulation of a CW laser leads to chirp-free and flat-top pulses or flaticons which exhibit a stable self-similar evolution. Upon collision, flaticons at different carrier frequencies, which may also occur in wavelength division multiplexed transmission systems, merge into a single, high-intensity, temporally and spatially localized rogue pulse.
Polarization modulation instability in a Manakov fiber system
2015
International audience; The Manakov model is the simplest multicomponent model of nonlinear wave theory: It describes elementary stable soliton propagation and multisoliton solutions, and it applies to nonlinear optics, hydrodynamics, and Bose-Einstein condensates. It is also of fundamental interest as an asymptotic model in the context of the widely used wavelength-division-multiplexed optical fiber transmission systems. However, although its physical relevance was confirmed by the experimental observation of Manakov (vector) solitons in a planar waveguide in 1996, there have in fact been no quantitative experiments confirming its validity for nonlinear dynamics other than soliton formatio…
From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N − 2 parameters
2017
International audience; We have already constructed solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants and wronskians of order 2N. These solutions have been called solutions of order N and they depend on 2N −1 parameters. We construct here N-order rational solutions. We prove that they can be written as a quotient of 2 polynomials of degree 2N(N +1) in x, y and t 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.
6-th order rational solutions to the KPI equation depending on 10 parameters
2017
International audience; Here we constuct rational solutions of order 6 to the Kadomtsev-Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 84 in x, y and t depending on 10 parameters. We verify that the maximum of modulus of these solutions at order 6 is equal to 2(2N + 1)2 = 338. We study the patterns of their modulus in the plane (x, y) and their evolution according time and parameters a1, a2, a3, a4, a5, b1, b2, b3, b4, b5. When these parameters grow, triangle and rings structures are obtained.