Exact solutions to the modified Korteweg-de Vries equation

A formula for certain exact solutions to the modified Korteweg-de Vries (mKdV) equation is obtained via the inverse scattering transform method. The kernel of the relevant Marchenko integral equation is written with the help of matrix exponentials as $$\Omega(x+y;t)=Ce^{-(x+y)A}e^{8A^3 t}B,$$ where the real matrix triplet $(A,B,C)$ consists of a constant $p\times p$ matrix $A$ with eigenvalues having positive real parts, a constant $p\times 1$ matrix $B$, and a constant $1\times p$ matrix $C$ for a positive integer $p$. Using separation of variables, the Marchenko integral equation is explicitly solved yielding exact solutions to the mKdV equation. These solutions are constructed in terms of the unique solution $P$ to the Sylvester equation $AP+PA=BC$ or in terms of the unique solutions $Q$ and $N$ to the respective Lyapunov equations $A^\dagger Q+QA=C^\dagger C$ and $AN+NA^\dagger=BB^\dagger$, where the $\dagger$ denotes the matrix conjugate transpose. Two interesting examples are provided.Comment: 15 pages, 1 figur

Symmetries for exact solutions to the nonlinear Schr\"odinger equation

A certain symmetry is exploited in expressing exact solutions to the focusing nonlinear Schr\"odinger 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 Schr\"odinger equation of any matrix size

Particle swarm optimization of GaAs-AlGaAS nanowire photonic crystals as two-dimensional diffraction gratings for light trapping

Semiconductor nanowire ordered arrays represent a class of bi-dimensional photonic crystals that can be engineered to obtain functional metamaterials. Here is proposed a novel approach, based on a particle swarm optimization algorithm, for using such a photonic crystal concept to design a semiconductor nanowire-based two-dimensional diffraction grating able to guarantee an in-plane coupling for light trapping. The method takes into account the experimental constraints associated to the bottom-up growth of nanowire arrays, by processing as input dataset all relevant geometrical and morphological features of the array, and returns as output the optimised set of parameters according to the desired electromagnetic functionality of the metamaterial. A case of study based on an array of tapered GaAs-AlGaAs core-shell nanowire heterostructures is discussed

Heisenberg ferromagnetism as an evolution of a spherical indicatrix: localized solutions and elliptic dispersionless reduction

A geometrical formulation of Heisenberg ferromagnetism as an evolution of a curve on the unit sphere in terms of intrinsic variables is provided and investigated. Given a vortex filament moving in an incompressible Euler fluid with constant density (under the local induction approximation hypotheses), the solutions of the classical Heisenberg ferromagnet equation are represented by the corresponding spherical (or tangent) indicatrix. The equations for the time evolution of the indicatrix on the unit sphere are given explicitly in terms of two intrinsic variables, the geodesic curvature and the arc-length of the curve. Notably, by considering the evolution with respect to slow variables and neglecting the dispersive terms, a novel elliptic dispersionless reduction of the Heisenberg ferromagnet model is obtained. The length of the spherical indicatrix is proved not to be conserved. Finally, a totally explicit algorithm is provided, allowing to construct a solution of the Heisenberg ferromagnet equation from a solution of Nonlinear Schrodinger equation, and, remarkably, viceversa, allowing to construct a solution of Nonlinear Schrodinger equation from a solution of the Heisenberg ferromagnet equation. As expected from the Zakharov-Takhtajan gauge equivalence, in the reflectionless case such a two-way map between solutions is shown to preserve the Inverse Scattering Transform spectra, and thus the localization

Quaternion algebra approach to nonlinear Schrödinger equations with nonvanishing boundary conditions

In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices

Surface Nano-Patterning for the Bottom-Up Growth of III-V Semiconductor Nanowire Ordered Arrays

Ordered arrays of vertically aligned semiconductor nanowires are regarded as promising candidates for the realization of all-dielectric metamaterials, artificial electromagnetic materials, whose properties can be engineered to enable new functions and enhanced device performances with respect to naturally existing materials. In this review we account for the recent progresses in substrate nanopatterning methods, strategies and approaches that overall constitute the preliminary step towards the bottom-up growth of arrays of vertically aligned semiconductor nanowires with a controlled location, size and morphology of each nanowire. While we focus specifically on III-V semiconductor nanowires, several concepts, mechanisms and conclusions reported in the manuscript can be invoked and are valid also for different nanowire materials

A Classical molecular dynamics study of recombination reactions in a microporous solid

Classical molecular dynamics calculations have been applied to the study of the recombination reaction of photodissociated radical species. Within a simplified reaction scheme it has been possible to get qualitative information about the influence of the environment. A comparison has been made between reactions in a liquid solvent and in a complex structured environment, such as a microporous silicate. Marked differences in the recombination yield and in the energy relaxation mechanism have been observed