Variant-Coherence Gaussian Sources

The celebrated Gaussian Schell model source with its shift-invariant degree of coherence may be the basis for devising sources with space-variant properties in the spirit of structured coherence. Starting from superpositions of Gaussian Schell model sources, we present two classes of genuine cross-spectral densities whose degree of coherence varies across the source area. The first class is based on the use of the Laplace transform while the second deals with cross-spectral densities that are shape-invariant upon paraxial propagation. For the latter, we present a set of shape-invariant cross-spectral densities for which the modal expansion can be explicitly found. We finally solve the problem of ascertain whether an assigned cross-spectral density is shape-invariant by checking if it satisfies a simple differential constraint.

Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System

The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.

Asymptotic variational analysis of incompressible elastic strings

Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

ALGORITHM FOR CONSTRUCTING A LAX PAIR AND A RECURSION OPERATOR FOR INTEGRABLE EQUATIONS

The method of constructing particular solutions to nonlinear partial differential equations based on the notion of differential constraint (or invariant manifold) is well known in the literature, see (Yanenko, 1961; Sidorov et al., 1984). The matter of the method is to add a compatible equation to a given equation and as a rule, the compatible equation is simpler. Such technique allows one to find particular solutions to a studied equation. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) there was proposed a scheme for constructing the Lax pairs and recursion operators for integrable partial differential equations based on the use of similar idea. A suitable generalization is to impose a differential constraint not on the equation, but on its linearization. The resulting equation is referred to as a generalized invariant manifold. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) it is shown that generalized invariant varieties allow efficient construction of Lax pairs and recursion operators of integrable equations. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

On a differential constraint in the continuum theory of growing solids

Предлагается один общий принцип постановки граничных условий в краевых задачах механики растущих тел. При выводе определяющих соотношений на поверхности наращивания используется аппарат алгебры рациональных инвариантов. Проведен вывод различных вариантов физически непротиворечивых дифференциальных ограничений на поверхности наращивания. Полученные условия справедливы для весьма широкого круга материалов и метаматериалов. Для использования сформулированных дифференциальных ограничений в конкретных приложениях необходима их экспериментальная идентификация. По этой причине полученные результаты могут служить общей основой в прикладных исследованиях по механике растущих тел.

Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed 𝓐-p quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone of 𝓐.