Sandra Carillo

The abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links

Explicit computation for a Kawamoto‐type equation shows that there is a rich associated symmetry structure for four separate hierarchies of nonlinear integrodifferential equations. Contrary to the general belief that symmetry groups for nonlinear evolution equations in 1+1 dimensions have to be Abelian, it is shown that, in this case, the symmetry group is noncommutative. Its semisimple part is isomorphic to the affine Lie algebra A(1)1 associated to sl(2,C). In two of the additional hierarchies that were found, an explicit dependence of the independent variable occurs. Surprisingly, the generic invariance for the Kawamoto‐type equation obtained in Rogers and Carillo [Phys. Scr. 36, 865 (1987)] via a reciprocal link to the Mobius invariance of the singularity equation of the Kaup–Kupershmidt (KK) equation only holds for one of the additional hierarchies of symmetry groups. Thus the generic invariance is not a universal property for the complete symmetry group of equations obtained by reciprocal links. In ...

Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity

Abstract Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared.

Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods

Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Backlund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a “Backlund chart” depicts links relating, in particular, the Korteweg–de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Backlund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework.

The action-angle transformation for soliton equations

For third order completely integrable equations in 1 + 1 dimensions canonical transformations which map (in the multi-soliton case) gradients of action variables to gradients of angle variables are constructed. Furthermore, this action-angle map is shown always to be an infinitesimal symmetry group generator of the interacting soliton equation.

Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions

Backlund transformations between all known completely integrable third-order differential equations in (1 + 1)-dimensions are established and the corresponding transformations formulas for their hereditary operators and Hamiltonian formulations are exhibited. Some of these Backlund transformations are not injective; therefore additional non-commutative symmetry groups are found for some equations. These non-commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra A(1)1. New completely integrable third-order integro-differential equations, some depending explicitly on x, are given. These new equations give rise to nonin equation. Connections between the singularity equations (from the Painleve analysis) and the nonlinear equations for interacting solitons are established. A common approach to singularity analysis and soliton structure is introduced. The Painleve analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine-Gordon and nonlinear Schrodinger equation. A method to recover the Painleve series from its constant level term is exhibit. The soliton-singularity transform is recognized to be connected to the Mobius group. This gives rise to a Darboux-like result for the spectral properties of the recursion operator. These connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons. Furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton-singularity transform. This then leads to the result that only for scaling degrees α = -1 and α = -2 the usual Painleve analysis can be carried out. A new invariance principle, connected to kernels of differential operators is discovered. This new invariance, for example, connects the explicit solutions of the Liouville equation with the Miura transform. Simple methods are exhibited which allow to compute out of N-soliton solutions of the KdV (Bargman potentials) explicit solutions of equations like the Harry Dym equation. Certain solutions are plotted.

Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions

The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold [“Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods,” J. Math. Phys. 50, 073510 (2009)]10.1063/1.3155080. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies.

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 Backlund chart, generalizing ...

ON THE RECURSION OPERATOR FOR THE NONCOMMUTATIVE BURGERS HIERARCHY

The noncommutative Burgers recursion operator is constructed via the Cole–Hopf transformation, and its structural properties are studied. In particular, a direct proof of its hereditary property is given.

Existence, uniqueness and exponential decay: An evolution problem in heat conduction with memory

A rigid linear heat conductor with memory conductor is considered. An evolution problem which arises in studying the thermodynamical state of the material with memory is considered. Specically, the time evolution of the temperature distribution within a rigid heat conductor with memory is investigated. The constitutive equations which characterize heat conduction with memory, involve an integral term since the temperature’s time derivative is connected to the heat ux gradient. The integro-dierenti al problem, when initial and boundary conditions are assigned, is studied to obtain existence and uniqueness results. Key tools, turn out to be represented by suitable expressions of the minimum free energy which allow to construct functional spaces which are both meaningful under the physical as well as the analytic viewpoint since therein the existence and uniqueness results can be established. Finally, conditions which guarantee exponential decay at innity are obtained.

A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity

We prove the existence and uniqueness of solution to a one-dimensional hyperbolic–parabolic system arising in the study of magneto-viscoelasticity. Specifically, the local existence and uniqueness result is proved on application of the fixed point theorem; then, a uniform a priori estimate of the solution is established. The latter, via a continuation method, allows as to obtain the global result. A crucial tool to achieve such a result is a technical lemma concerning the only viscoelastic contribution; it relies on the assumptions that the memory kernel is positive, monotonically non-increasing and convex.