Search results for "Riemann theta function"
showing 8 items of 8 documents
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.
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.
Six-parameters deformations of fourth order Peregrine breather solutions of the NLS equation.
2013
We construct solutions of the focusing NLS equation as a quotient of two determinants. This formulation gives in the case of the order 4, new deformations of the Peregrine breather with 6 real parameters. We construct families of quasi-rational solutions of the NLS equation and describe the apparition of multi rogue waves. With this method, we construct the analytical expressions of deformations of the Peregrine breather of order 4 with 6 real parameters and plot different types of rogue waves.
Degenerate determinant representation of solutions of the NLS equation, higher Peregrine breathers and multi-rogue waves.
2012
We present a new representation of solutions of the focusing NLS equation as a quotient of two determinants. This work is based on a recent paper in which we have constructed a multi-parametric family of this equation in terms of wronskians. This formulation was written in terms of a limit involving a parameter. Here we give a very compact formulation without presence of a limit. This is a completely new result which gives a very efficient procedure to construct families of quasi-rational solutions of the NLS equation. With this method, we construct Peregrine breathers of orders N=4 to 7 and multi-rogue waves associated by deformation of parameters.
Families of quasi-rational solutions of the NLS equation as an extension of higher order Peregrine breathers.
2011
We construct a multi-parametric family of solutions of the focusing NLS equation from the known result 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 get 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 to get a simpler derivation of the generic formulas corres…
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.
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.
Determinant representation of NLS equation, Ninth Peregrine breather and multi-rogue waves
2012
This article 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 its deformations. Here we construct Peregrine breathers of order $N=9$ and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.