Francesco Demontis

Exact solutions to the focusing nonlinear Schrödinger 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.

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

Explicit solutions of the cubic matrix nonlinear Schrödinger equation

x

The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

and the temporal variable

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

t,

Symmetries for exact solutions to the nonlinear Schrödinger equation

and they are exponentially asymptotic to integer multiples of

Exact Solutions to the Nonlinear Schrödinger Equation

2\pi

An exact macroscopic extended model with many moments for ultrarelativistic gases

as

Wave speeds in the macroscopic extended model for ultrarelativistic gases

x\to\pm\infty.

Effective Generation of Closed-form Soliton Solutions of the Continuous Classical Heisenberg Ferromagnet Equation

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.