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Sara Lombardo

Publications and source records attributed to Sara Lombardo.

18 recordsLinked to original sources
arXiv2025

Normal forms of elliptic automorphic Lie algebras and Landau-Lifshitz type of equations

We present normal forms of elliptic automorphic Lie algebras with dihedral symmetry of order 4, which arise naturally in the context of Landau-Lifshitz type of equations. These normal forms provide a transparent description and allow a classification of such Lie algebras over $\mathbb{C}$. Using this perspective, we show that a Lie algebra introduced by Uglov, as well as the hidden symmetry algebra of the Landau-Lifshitz equation by Holod, are both isomorphic to an elliptic $\mathfrak{sl}(2,\mathbb{C})$-current algebra. Furthermore, we realise the Wahlquist-Estabrook algebra of the Landau-Lifshitz equation in terms of elliptic automorphic Lie algebras. This construction reveals that, as complex Lie algebras, it is isomorphic to the direct sum of an $\mathfrak{sl}(2,\mathbb{C})$-current algebra and the two-dimensional abelian Lie algebra $\mathbb{C}^2$. Finally, we explicitly implement the automorphic Lie algebra framework in the context of an $n$-component generalisation of the Landau-Lifshitz equation by Golubchik and Sokolov in the case of $n=3$.

math.RA↗
arXiv2024

A classification of automorphic Lie algebras on complex tori

We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$. For each case we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2(\mathbb{Z})$, apart from four cases, which are all isomorphic to Onsager's algebra.

math.RA↗
arXiv2022

Periodic and solitary wave solutions of the long wave-short wave Yajima-Oikawa-Newell model

Models describing long wave-short wave resonant interactions have many physical applications from fluid dynamics to plasma physics. We consider here the Yajima-Oikawa-Newell (YON) model, which has been recently introduced combining the interaction terms of two long wave-short wave, integrable models, one proposed by Yajima-Oikawa, and the other one by Newell. The new YON model contains two arbitrary coupling constants and it is still integrable - in the sense of possessing a Lax pair - for any values of these coupling constants. It reduces to the Yajima-Oikawa or the Newell systems for special choices of these two parameters. We construct families of periodic and solitary wave solutions, which display the generation of very long waves. We also compute the explicit expression of a number of conservation laws.

nlin.SI↗
arXiv2021

A new integrable model of long wave-short wave interaction and linear stability spectra

We consider the propagation of short waves which generate waves of much longer (infinite) wave-length. Model equations of such long wave-short wave resonant interaction, including integrable ones, are well-known and have received much attention because of their appearance in various physical contexts, particularly fluid dynamics and plasma physics. Here we introduce a new long wave-short wave integrable model which generalises those first proposed by Yajima-Oikawa and by Newell. By means of its associated Lax pair, we carry out the linear stability analysis of its continuous wave solutions by introducing the stability spectrum as an algebraic curve in the complex plane. This is done starting from the construction of the eigenfunctions of the linearised long wave-short wave model equations. The geometrical features of this spectrum are related to the stability/instability properties of the solution under scrutiny. Stability spectra for the plane wave solutions are fully classified in the parameter space together with types of modulational instabilities.

nlin.SI↗
arXiv2020

Wild Local Structures of Automorphic Lie Algebras

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

math.RT↗
arXiv2020

Automorphic Lie Algebras and Cohomology of Root Systems

A cohomology theory of root systems emerges naturally in the context of Automorphic Lie Algebras, where it helps formulating some structure theory questions. In particular, one can find concrete models for an Automorphic Lie Algebra by integrating cocycles. In this paper we define this cohomology and show its connection with the theory of Automorphic Lie Algebras. Furthermore, we discuss its properties: we define the cup product, we show that it can be restricted to symmetric forms, that it is equivariant with respect to the automorphism group of the root system, and finally we show acyclicity at dimension two of the symmetric part, which is exactly what is needed to find concrete models for Automorphic Lie Algebras. Furthermore, we show how the cohomology of root systems finds application beyond the theory of Automorphic Lie Algebras by applying it to the theory of contractions and filtrations of Lie algebras. In particular, we show that contractions associated to Cartan $\mathbb{Z}$-filtrations of simple Lie algebras are classified by $2$-cocycles, due again to the vanishing of the symmetric part of the second cohomology group.

math.RA↗
arXiv2019

Hereditary Automorphic Lie Algebras

We show that Automorphic Lie Algebras which contain a Cartan subalgebra with a constant spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianisation of the Automorphic Lie Algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite dimensional space of said equivariant vectors to a finite dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

math-ph↗
arXiv2018

Integrability and linear stability of nonlinear waves

It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation by using only their associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general $N \times N$ matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schroedinger system and the multi-wave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for $N = 3$ for the particular system of two coupled nonlinear Schroedinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.

nlin.SI↗
arXiv2015

Higher dimensional Automorphic Lie Algebras

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on one hand a powerful tool from the computational point of view, on the other it opens new questions from an algebraic perspective, which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that Automorphic Lie Algebras associated to the $\mathbb{T}\mathbb{O}\mathbb{Y}$ groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only, thus they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.

math-ph↗
arXiv2014

Automorphic Lie Algebras with dihedral symmetry

The concept of Automorphic Lie Algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever-Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is $\mathfrak{sl}_2(\mathbb{C})$ and the poles of the Automorphic Lie Algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In the present research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of Automorphic Lie Algebras with dihedral symmetry, valid for poles at exceptional and generic orbits.

nlin.SI↗
arXiv2014

Vector Rogue Waves and Baseband Modulation Instability in the Defocusing Regime

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multi-component systems, from oceanography to optics and plasma physics.

nlin.SI↗
arXiv2013

Rational solitons of wave resonant interaction models

Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector Nonlinear Schroedinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic construction relies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rational (mixed rational-exponential) solutions and find, for the first time to our knowledge, a broad family of such solutions of the three wave resonant interaction equations.

nlin.SI↗
arXiv2013

Rogue waves emerging from the resonant interaction of three waves

We introduce a novel family of analytic solutions of the three-wave resonant interaction equations to the purpose of modeling unique events, i.e. "amplitude peaks" which are isolated in space and time. The description of these solutions is likely to be a crucial step in the understanding and forecasting of rogue-waves in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.

nlin.SI↗
arXiv2009

On the Classification of Automorphic Lie Algebras

It is shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl2-Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.

nlin.SI↗
arXiv2009

Multicomponent integrable wave equations II: Soliton solutions

The Darboux Dressing Transformations developed in our previous paper (Multicomponent integrable wave equations I. Darboux-Dressing Transformation, J. Phys. A: Math. Theor. 40, 961-977, 2007) are here applied to construct soliton solutions for a class of boomeronic type equations. The vacuum (i.e. vanishing) solution and the generic plane wave solution are both dressed to yield one soliton solutions. The formulae are specialised to the particularly interesting case of the resonant interaction of three waves, a well-known model which is of boomeronic type. For this equation a novel solution which describes three locked dark pulses (simulton) is introduced.

nlin.SI↗
arXiv2006

Multicomponent integrable wave equations I. Darboux-Dressing Transformation

The Darboux-Dressing Transformations are applied to the Lax pair associated to systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both bright and dark soliton solutions. The general formalism is set up and the relevant equations are explicitly solved. Several instances of multicomponent wave equations of applicative interest, such as vector nonlinear Schroedinger-type equations and three resonant wave equations are considered.

nlin.SI↗
arXiv2005

Exact solutions of the 3-wave resonant interaction equation

The Darboux--Dressing Transformations are applied to the Lax pair associated to the system of nonlinear equations describing the resonant interaction of three waves in 1+1 dimensions. We display explicit solutions featuring localized waves whose profile vanishes at the spacial boundary plus and minus infinity, and which are not pure soliton solutions. These solutions depend on an arbitrary function and allow to deal with collisions of waves with various profiles.

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