Search results for "Mathematical physics"
showing 10 items of 2687 documents
A Symplectic Kovacic's Algorithm in Dimension 4
2018
Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions $X$ satisfies $X^t J X=J$ where $J$ is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if $L$ is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order $4$. Moreover, using Klein's Theorem, algebraic solution…
Can there be a general nonlinear PDE theory for existence of solutions ?
2010
Updated version of the 2004 paper arxiv:math/0407026; Contrary to widespread perception, there is ever since 1994 a unified, general type independent theory for the existence of solutions for very large classes of nonlinear systems of PDEs. This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs. The method can also deal with associated initial and/or boundary value problems. The solutions obtained can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains. It is important to note that the use of the or…
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
Remembering Ludwig Dmitrievich Faddeev, Our Lifelong Partner in Mathematical Physics
2018
International audience; We briefly recount the long friendship that developed between Ludwig and us (Moshé Flato and I), since we first met at ICM 1966 in Moscow. That friendship extended to his school and family, and persists to this day. Its strong personal impact and main scientific components are sketched, including reflections on what mathematical physics is (or should be).
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…
An attempt to classification of the quasi rational solutions to the NLS equation
2015
Based on a representation in terms of determinants of order 2N , an attempt to classification of quasi rational solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) is given and several conjectures about the structure of the solutions are also formulated. These solutions 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 depending on 2N −2 parameters. It is remarkable to mention that in this representation, when all parameters are equal to 0, we recover the PN breathers.
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.
N-order rational solutions to the Johnson equation depending on 2N - 2 parameter
2017
International audience; 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 2N(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.
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.