arXiv subjects

A. S. Carstea

Publications and source records attributed to A. S. Carstea.

17 recordsLinked to original sources

arXiv2025

Perturbation of Traveling Boussinesq Solitons by Periodic Bathymetry

We investigate the perturbations induced by a periodic bathymetry on traveling Boussinesq solitons in a two-dimensional configuration. We present two perturbation approaches to solve the nonlinear, dispersive and non-autonomous differential equations of the model and compare the solutions with numerical simulations of the original system of equations. In the approximation for small periodic corrugations we built the solutions as modulated traveling waves using Fourier series. The coefficients of the series are solved using the Green function method and the pathordered exponential method. At the second order in the relative height of bed corrugations, we obtain the perturbation as the fourth-order linear dispersive waves generated by the modulated traveling soliton in its wake. In the second approach, we rewrite the Boussinesq system into a perturbed Korteweg-de Vries (KdV) nonlinear equation, and obtain the corresponding perturbed solitons. These analytic solutions are compared with the results of numerical simulations, for various parameters that characterize the effects of nonlinearity, dispersion, and bottom bathymetry. We also discuss the stability of the perturbed solitons in time. The perturbation approaches developed in this study are valid for any type of periodic bathymetries, and the method can readily be extended to non-periodic ones.

nlin.PS↗

arXiv2025

Model for self-organized Leidenfrost rotating polygons as cnoidal waves

The remarkable appearance of self-organized regular and peaked polygonal rotating patterns in shallow Leidenfrost rings is investigated as a balance between surface tension geometry and nonlinear terms of Euler equation. Using the Boussinesq shallow convection approximation and a specialized expansion of the Laplace equation solutions, we derive a nonlinear equation that can be integrated in terms of elliptic functions. The model rigorously accounts for surface tension, the contribution of the poloidal rolling vortex, and the interplay between buoyancy-driven and thermocapillary flows. We obtain cnoidal waves solutions describing the dynamics of the inner free surface of the Leidenfrost ring, to predict polygonal patterns in liquid nitrogen. These predictions are compared with experimental observations.Additionally, we introduce a simplified model based on poloidal averaging of the capillary pressure, leading to a Korteweg-de Vries-type equation. This simplified model not only reproduces the cnoidal wave solutions but also predicts new trigonometric solutions, offering insights into the formation of peaked polygons.

physics.flu-dyn↗

arXiv2021

Nonlinear Schrodinger Equation Solitons on Quantum Droplets

Irrotational ow of a spherical thin liquid layer surrounding a rigid core is described using the defocusing nonlinear Schrodinger equation. Accordingly, azimuthal moving nonlinear waves are modeled by periodic dark solitons expressed by elliptic functions. In the quantum regime the algebraic Bethe ansatz is used in order to capture the energy levels of such motions, which we expect to be relevant for the dynamics of the nuclear clusters in deformed heavy nuclei surface modeled by quantum liquid drops. In order to validate the model we match our theoretical energy spectra with experimental results on energy, angular momentum and parity for alpha particle clustering nuclei.

nlin.PS↗

arXiv2019

Super-QRT and 4D-mappings reduced from the lattice super-KdV equation

Starting from the complete integrable lattice super-KdV equation, two super-mappings are obtained by performing a travelling-wave reduction. The first one is linear and the second is a four dimensional super-QRT mapping containing both Grassmann commuting and anti-commuting dependent variables. Adapting the classical "staircase" method to the Lax super-matrices of the lattice super-KdV equation, we compute the Lax super-matrices of the mapping and the two invariants; the first one is a pure nilpotent commuting quantity and the second one is given by an elliptic curve containing nilpotent commuting Grassmann coefficients as well. In the case of finitely generated Grassmann algebra with two generators, the super-QRT mapping becomes a four-dimensional ordinary discrete dynamical system that has two invariants but does not satisfy singularity confinement criterion. It is also observed that the dynamical degree of this system grows quadratically.

nlin.SI↗

arXiv2017

Bilinear approach to Kuperschmidt super-KdV type equations

Hirota bilinear form and soliton solutions for super-KdV of Kuperschmidt (Kuper-KdV) are given. It is shown that even though the collision of supersolitons is more complicated than in the case of supersymmetric KdV of Manin-Radul, the asymptotic effect of the interaction is simpler. As a physical application it is shown that the well known FPU problem having a phonon-mediated interaction of some internal degrees of freedom expressed through grassmann fields, goes to the Kuper-KdV equation in a multiple-scale approach.

nlin.SI↗

arXiv2017

Coupled Ablowitz-Ladik equations with branched dispersion

Complete integrability and multisoliton solutions are discussed for a multicomponent Ablowitz-Ladik system with branched dispersion relation. It is also shown that starting from a "diagonal" (in two-dimensions) completely integrable Ablowitz-Ladik equation, one can obtain all the results using a periodic reduction.

nlin.SI↗

arXiv2017

Bilinear approach to the supersymmetric Gardner equation

We study a supersymmetric version of the Gardner equation (both focusing and defocusing) using the superbilinear formalism. This equation is new and cannot be obtained from supersymmetric modified Korteweg-de Vries equation with a nonzero boundary condition. We construct supersymmetric solitons and then by passing to the long-wave limit in the focusing case obtain rational nonsingular solutions. We also discuss the supersymmetric version of the defocusing equation and the dynamics of its solutions.

math-ph↗

arXiv2015

Alternative integrable discretisation of Korteweg de Vries equation

We present an alternative integrable discretization of differential-difference KdV equation based on Hirota bilinear formalism. It is shown that using two tau functions the direct discretisation of the bilinear equations gives immediately the well known discrete KdV equation. We comment also on integrability and relation with the classical bilinear form involving only one tau function.

nlin.SI↗

arXiv2015

On some new forms of lattice integrable equations

Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon - type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.

nlin.SI↗

arXiv2015

On various integrable discretisations of a general two component Volterra system

We present two integrable discretisations of a general differential-difference bicomponent Volterra system. The results are obtained by discretising directly the corresponding Hirota bilinear equations in two different ways. Multisoliton solutions are presented together with a new discrete form of Lotka-Volterra equation obtained by an alternative bilinearisation.

nlin.SI↗

arXiv2014

Multisoliton solution and super-bilinear form of lattice supersymmetric KdV equation

Hirota bilinear form and multisoliton solution for semidiscrete and fully discrete (difference-difference) versions of supersymmetric KdV equation found by Xue, Levi and Liu [1] is presented. The solitonic interaction term displays a fermionic dressing factor as in the continuous supersymmetric case. Using bilinear equations it is shown also that there can be constructed a new integrable semidiscrete (and fully discrete) version of supersymmetric KdV which has simpler bilinear form but more complicated interaction dressing. Its continuum limit is also computed.

nlin.SI↗

arXiv2012

A classification of two dimensional integrable mappings and rational elliptic surfaces

We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which exchange fibers. We also show an equivalent condition when a generalized Halphen surface becomes a Halphen surface of index m.

math.DS↗

arXiv2006

Proteomic nonlinear waves in networks of transcriptional regulators

A chain of connected genes with activation-repression links is analysed. It is shown that for various promoter activity functions (parametrised by Hill coefficient) the equations describing the concentrations of transcription factors, are differential-difference KdV-type with perturbations. In the case of large Hill coefficient the proteomic signal along the gene network is given by a superposition of perturbed dark solitons of defocusing differential-difference mKdV equation. Biological implications are discussed.

q-bio.GN↗

arXiv2004

Localised and nonlocalised structures in nonlinear lattices with fermions

We discuss the quasiclassical approximation for the equations of motions of a nonlinear chain of phonons and electrons having phonon mediated hopping. Describing the phonons and electrons as even and odd grassmannian functions and using the continuum limit we show that the equations of motions lead to a Zakharov-like system for bosonic and fermionic fields. Localised and nonlocalised solutions are discussed using the Hirota bilinear formalism. Nonlocalised solutions turn out to appear naturally for any choice of wave parameters. The bosonic localised solution has a fermionic dressing while the fermionic one is an oscillatory localised field. They appear only if some constraints on the dispersion are imposed. In this case the density of fermions is a strongly localised travelling wave. Also it is shown that in the multiple scales approach the emergent equation is linear. Only for the resonant case we get a nonlinear fermionic Yajima-Oikawa system. Physical implications are discussed.

nlin.SI↗

arXiv2000

Hirota bilinear formalism and Supersymmetry

Extending the gauge-invariance principle for $τ$ functions of the standard bilinear formalism to the supersymmetric case, we define ${\cal N}=1$ supersymmetric Hirota bilinear operators. Using them we bilinearize supersymmetric nonlinear evolution equations. The super-soliton solutions are discussed. As a quite strange paradox it is shown that the Lax integrable supersymmetric KdV of Manin-Radul-Mathieu equation does not possesses N super-soliton solution for $N\geq 3$ for arbitrary parameters. Only for a particular choice of them the N super-soliton solution exists.

nlin.SI↗

arXiv1999

Beyond Nonlinear Schrödinger Equation Approximation for an Anharmonic Chain with Harmonic Long Range Interaction

Multi scales method is used to analyze a nonlinear differential-difference equation. In order $ε^3$ the NLS equation is found to determine the space-time evolution of the leading amplitude. In the next order this has to satisfy a complex mKdV equation (the next in the NLS hierarchy) in order to eliminate secular terms. The zero dispersion point case is also analyzed and the relevant equation is a modified NLS equation with a third order derivative term included

solv-int↗

arXiv1998

Extension of the bilinear formalism to supersymmetric KdV-type equations

Extending the gauge-invariance principle for τfunctions of the standard bilinear formalism to the supersymmetric case, we define N=1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV, Sawada-Kotera, Hirota-Satsuma). The solutions for multiple collisions of super-solitons and extension to SUSY sine-Gordon are also discussed.

solv-int↗