arXiv subjects
Publications and source records attributed to Sandra Carillo.
A class of moving boundary problems of Stefan-type for an extension of the 2+1- dimensional Gardner equation is shown to be amenable to exact solution by application of a Ermakov- Painlev\'e II symmetry reduction.
A reciprocal transformation is applied to link an integrable extension of the classical solitonic mKdV to a novel nonlinear evolution equation incorporating a source term. Application of Ermakov-Painleve' II symmetry reduction is made to determine the exact solution to a class of associated moving boundary problems of Stefan-type.
Here classes of moving boundary problems of Stefan-type for both an established non-linear evolution equation of cuspon theory and novel reciprocally linked solitonic equations are shown to be solvable via Painleve' II symmetry reduction.
The term material with memory is generally used to indicate materials whose mechanical and/or thermodynamical behaviour depends not only on the process at the present time but also on the history of the process itself. Crucial in heat conductors with memory is the heat relaxation function which models the thermal response of the material. The present study is concerned about a thermodynamical problem with memory "aging"; that is, we analyze the temperature evolution within a rigid heat conductor with memory whose relaxation function takes into account the aging of the material. In particular, we account for variations of the relaxation function due to a possible deterioration of the thermal response of the material related to its age.
The goal of this note is to study nonlinear parabolic problems nonlocal in time and space. We first establish the existence of a solution and its uniqueness in certain cases. Finally we consider its asymptotic behaviour.
A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on proving a new invariance admitted by a third order nonlinear evolution equation and, on the other one, on the construction of solutions. Indeed, via B\"acklund transformations, a {\it B\"acklund chart}, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide B\"acklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the {\it Korteweg-deVries interacting soliton} (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian {\it B\"acklund chart}, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.
Assuming Newton's law of cooling, the propagation and structure of isothermal acoustic shocks are studied under four different viscosity laws. Employing both analytical and numerical methods, 1D traveling wave solutions for the velocity and density fields are derived and analyzed. For each viscosity law considered, expressions for both the shock thickness and the asymmetry metric are determined. And, to ensure that isothermal flow is achievable, upper bounds on the associated Mach number values are derived/computed using the isothermal version of the energy equation.
The role of Schwarzian derivative in the study of nonlinear ordinary differential equations is revisited. Solutions and invariances admitted by Painlev\'e XXV-Ermakov equation, Ermakov equation and third order linear equation in a normal form are shown to be based on solutions of the Schwarzian equation. Starting from the Riccati equation and the second order element of the Riccati chian as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of B\"acklund transformations which link the linear and nonlinear equations under investigation are obtained. Some examples with relevant applications are given and discussed.
The KdV eigenfunction equation is considered: some explicit solutions are constructed. These, to the best of the authors' knowledge, new solutions represent an example of the powerfulness of the method devised. Specifically, B\"acklund transformation are applied to reveal algebraic properties enjoyed by nonlinear evolution equations they connect. Indeed, B\"acklund transformations, well known to represent a key tool in the study of nonlinear evolution equations, are shown to allow the construction of a net of nonlinear links, termed "B\"acklund chart", connecting Abelian as well as non Abelian equations. The present study concerns third order nonlinear evolution equations which are all connected to the KdV equation. In particular, the Abelian wide B\"acklund chart connecting these nonlinear evolution equations is recalled. Then, the links, originally established in the case of Abelian equations, are shown to conserve their validity when non Abelian counterparts are considered. In addition, the non-commutative case reveals a richer structure related to the multiplicity of non-Abelian equations which correspond to the same Abelian one. Reduction from the nc to the commutative case allow to show the connection of the KdV equation with KdV eigenfunction equation, in the "scalar" case. Finally, recently obtained matrix solutions of the mKdV equations are recalled.
An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries equation. In: Nonlinear Dynamics of Structures, Systems and Devices, edited by W. Lacarbonara, B. Balachandran, J. Ma, J. Tenreiro Machado, G. Stepan (Springer, Cham, 2020), pp. 75-83). In fact, the solutions in Ref.7 are part of a subclass studied in detail by the authors in a forthcoming publication. Here several solutions beyond this subclass are constructed and discussed with respect to qualitative properties.
We investigate the possibility to obtain a thermal diode with functionally graded Si-Ge alloys. A wire with variable section is considered. After the introduction of a formula giving the thermal conductivity of the wire as a function of the species content and of the diameter of the wire, numerical and analytical results are presented supporting the impracticability to get a thermal diode with the characteristics here considered. However, the present study opens the way to further generalisations amenable to give applicative promising results.
A model problem of magneto-elastic body is considered. Specifically, the case of a two dimensional circular disk is studied. The functional which represents the magneto-elastic energy is introduced. Then, the minimisation problem, referring to the simplified two-dimensional model under investigation, is analysed. The existence of a minimiser is proved and its dependence on the eigenvalues of the problem is investigated. A bifurcation result is obtained corresponding to special values of the parameters.
We examine second-sound phenomena in a class of rigid, thermally conducting, solids that are described by a special case of the Maxwell--Catteneo flux law. Employing both analytical and numerical methods, we examine both temperature-rate waves and thermal traveling waves in this class of thermal conductor, which have recently been termed Graffi--Franchi--Straughan type conductors. In the present study, the temperature-dependent nature of the thermal relaxation time, which is the distinguishing feature of this class of conductors, gives rise to a variety of nonlinear effects; in particular, finite-time temperature-rate wave blow-up and temperature traveling waveforms which exhibit a "tongue". The presentation concludes with a discussion of possible follow-on studies.
Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equations. Matrix equation can be viewed as a specialisation of operator equations in the finite dimensional case when operators are finite dimensional and, hence, admit a matrix representation. Baecklund transformations allow to reveal structural properties [S. Carillo and C. Schiebold, J. Math. Phys. 50 (2009), 073510] enjoyed by non-commutative KdV- type equations, such as the existence of a recursion operator. Operator methods are briefly recalled aiming to show how they can be applied to construct soliton solutions. These methods, combined with Baecklund transformations, allow to obtain solutions of matrix soliton equations. Explicit solution formulae previously constructed [C. Schiebold, Glasgow Math. J. 51, 147-155 (2009)], [S. Carillo and C. Schiebold, J. Math. Phys. 52 (2011), 053507] are used to obtain 2 x 2 and 3 x 3 matrix mKdV solutions. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit
The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.
Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.
Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a singular kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional singular kernel initial boundary value problem.
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.