arXiv subjects

Sandra Carillo

Publications and source records attributed to Sandra Carillo.

26 records · Page 2Linked to original sources

arXiv2018

KdV-type equations linked via Baecklund transformations: remarks and perspectives

Third order nonlinear evolution equations, that is the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered: they all are connected via Baecklund transformations. These links can be depicted in a wide Baecklund Chart} which further extends the previous one constructed in [22]. In particular, the Baecklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Baecklund Chart. That is, such a Baecklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30 is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Baecklund chart.

math.AP↗

arXiv2018

The Gross-Pitaevskii equation: Bäcklund transformations and admitted solutions

Bäcklund transformations are applied to study the Gross-Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwartzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross-Pitaevskii equation are obtained.

math-ph↗

arXiv2018

Some remarks on the model of rigid heat conductor with memory: unbounded heat relaxation function

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t=0$. That is, it is represented by a regular integrable function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions on $k$, whenever the heat flux is the same also the related thermal work is the same. Thus, also in the case under investigation, the notion of equivalence is introduced and its physical relevance is pointed out.

math-ph↗

arXiv2017

Ermakov-Pinney and Emden-Fowler equations: new solutions from novel Bäcklund transformations

The class of nonlinear ordinary differential equations $y^{\prime\prime}y = F(z,y^2)$, where F is a smooth function, is studied. Various nonlinear ordinary differential equations, whose applicative importance is well known, belong to such a class of nonlinear ordinary differential equations. Indeed, the Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov equations are among them. Bäcklund transformations and auto Bäcklund transformations are constructed: these last transformations induce the construction of a ladder of new solutions adimitted by the given differential equations starting from a trivial solutions. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficulty to apply.

math-ph↗

arXiv2017

A novel noncommutative KdV-type equation, its recursion operator, and solitons

A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived.

math-ph↗

arXiv2016

Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart

Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.

math-ph↗

arXiv2016

Recursion Operators admitted by non-Abelian Burgers equations: Some Remarks

The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here con- sidered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as $r_t = r_{xx} + 2r_x r$. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be the hereditary. These results are proved via direct computations as well as via computer assisted manipulations; ad hoc routines are needed to treat non-Abelian quantities and relations among them. The obtained recursion operator generates the $mirror$ non-Abelian Burgers hierarchy. The latter, when the unknown operator $r$ is replaced by a real valued function reduces to the usual (commutative) Burgers hierarchy. Accordingly, also the recursion operator reduces to the usual Burgers one.

math.AP↗

arXiv2016

A magneto-viscoelasticity problem with a singular memory kernel

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analyzed, it is obtained coupling an integro-differential equation modeling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modeling the presence of a magnetic field. The case under investigation generalizes a previous study since the relaxation function is allowed to be unbounded at the origin, provided it belongs to $L^1$; the magnetic model equation adopted, as in the previous results [21,22, 24, 25] is the penalized Ginzburg-Landau magnetic evolution equation.

math.AP↗