Search results for "determinant"

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…

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.

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.

The so-called Reaction object construction: Reaction or co-predication?

2017

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.

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.

Response Power Spectrum of Multi-Degree-of-Freedom Nonlinear Systems by a Galerkin Technique

2003

This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation o…

Quantized ATDHF: theory and realistic applications to heavy ion fusion

1982

The quantized ATDHF theory is reviewed and discussed in the context of the generator coordinate method. This allows for a derivation which does not require an a posteriori quantization process. The ATDHF equations are then solved numerically on a coordinate and momentum grid in fully three dimensional geometry. The theory is applied to various heavy ion systems, where potentials, mass parameters and quantum corrections are evaluated and compared to conventional results from constrained Hartree-Fock. Subbarrier fusion cross sections are calculated and compared with experiment.

Kompleksi skaitļi. Determinanti. Alģebraiski nolīdzinājumi. Parciāldaļas: lekcijas lasītas Latvijas Universitātes Inženierzinātņu un mechanikas fakul…

1931

On the implementation of weno schemes for a class of polydisperse sedimentation models

2011

The sedimentation of a polydisperse suspension of small rigid spheres of the same density, but which belong to a finite number of species (size classes), can be described by a spatially one-dimensional system of first-order, nonlinear, strongly coupled conservation laws. The unknowns are the volume fractions (concentrations) of each species as functions of depth and time. Typical solutions, e.g. for batch settling in a column, include discontinuities (kinematic shocks) separating areas of different composition. The accurate numerical approximation of these solutions is a challenge since closed-form eigenvalues and eigenvectors of the flux Jacobian are usually not available, and the characte…