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Cornelis van der Mee

Publications and source records attributed to Cornelis van der Mee.

9 recordsLinked to original sources
arXiv2025

Matrix Zakharov-Shabat Systems with Zero Diagonal Entry

In this article we develop the direct and inverse scattering theory of the Ablowitz-Kaup-Newell-Segur (AKNS) system $\bv_x=(ik\zS+\CQ(x))\bv$, where $\zS$ is a diagonal $n\times n$ matrix with diagonal entries $1$ and $-1$ and a single zero diagonal entry and $\CQ(x)$ is an $n\times n$ potential anticommuting with $\zS$ with entries in $L^1(\R)$. We derive the time evolution of the scattering data which, through the inverse scattering transform, lead to the solution of the initial-value problem for a system of long-wave-short-wave equations.

math.FA↗
arXiv2021

From the AKNS system to the matrix Schroedinger equation with vanishing potentials: Direct and inverse problems

We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schroedinger equation. The corresponding Miura transformation which allows this connection, converts the focusing matrix nonlinear Schroedinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.

nlin.SI↗
arXiv2015

Scattering data computation for the Zakharov-Shabat system

A numerical method to solve the direct scattering problem for the Zakharov-Shabat system associated to the initial value problem for the nonlinear Schroedinger equation is proposed. The method involves the numerical solution of Volterra integral systems with structured kernels and the identification of coefficients and parameters appearing in monomial-exponential sums. Numerical experiments confirm the effectiveness of the proposed technique.

math.NA↗
arXiv2010

Exact Solutions to the Sine-Gordon Equation

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable $x$ and the temporal variable $t,$ and they are exponentially asymptotic to integer multiples of $2π$ as $x\to\pm\infty.$ The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.

nlin.SI↗
arXiv2009

Exact solutions to the nonlinear Schrödinger equation

A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schrödinger equation on the line with a cubic nonlinearity. With motivation by the inverse scattering transform and help from the state-space method, an explicit formula is obtained to express such exact solutions in a compact form in terms of a matrix triplet and by using matrix exponentials. Such solutions consist of multisolitons with any multiplicities, are analytic on the entire $xt$-plane, decay exponentially as $x\to\pm\infty$ at each fixed $t,$ and can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates $x$ and $t.$ Various equivalent forms of the matrix triplet are presented yielding the same exact solution.

nlin.SI↗
arXiv2009

A unified approach to Darboux transformations

We analyze a certain class of integral equations related to Marchenko equations and Gel'fand-Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to Darboux transformations associated with various systems of ordinary differential operators. We illustrate our theory by deriving the Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a discrete eigenvalue is added to the spectrum.

nlin.SI↗
arXiv2009

Symmetries for exact solutions to the nonlinear Schrödinger equation

A certain symmetry is exploited in expressing exact solutions to the focusing nonlinear Schrödinger equation in terms of a triplet of constant matrices. Consequently, for any number of bound states with any number of multiplicities the corresponding soliton solutions are explicitly written in a compact form in terms of a matrix triplet. Conversely, from such a soliton solution the corresponding transmission coefficients, bound-state poles, bound-state norming constants and Jost solutions for the associated Zakharov-Shabat system are evaluated explicitly. It is also shown that these results hold for the matrix nonlinear Schrödinger equation of any matrix size.

nlin.SI↗
arXiv2007

Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.

nlin.SI↗