Saadet S. Özer | Searcharxiv

arXiv2026

Equivalence Groups of Inhomogeneous Hyperelasticity

Equivalence groups associated with the field equations of arbitrary motions of inhomogeneous hyperelastic solids are investigated. The explicit forms of the infinitesimal generators, namely the components of the isovector field associated with the equivalence groups are derived. Additional constraints on the isovector components are established to ensure that the transformed equations are supported by a strain energy function. Possible mappings between the equations of materials with different properties are examined. Not only the transformations between homogeneous and inhomogeneous hyperelasticity equations, but also the transformations between the equations and strain energy functions of isotropic and anisotropic materials are investigated. The results are illustrated with several examples.

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arXiv2026

Wave Selection at an $O(2)$-Hopf Bifurcation in Conservative Two-Component PDE Systems

At an $O(2)$-equivariant Hopf bifurcation a spatially periodic system selects between traveling waves and standing waves, and which branch appears and whether it is stable is decided by two cubic normal form coefficients. Obtaining these coefficients for a given PDE has required a derivation carried out afresh for each model. We remove that step for two-component systems with conservative polynomial differential nonlinearities on a one-dimensional periodic domain, deriving closed form formulas for both coefficients. They are expressed directly in terms of the array of nonlinear PDE coefficients and the spectral data of the linearization, its critical eigenvectors, adjoint eigenvectors, and non-critical resolvents. We give verifiable conditions under which the underlying center manifold reduction is valid: a structural condition on the principal part of the linearization that yields the required resolvent estimate, and a condition on the Fourier symbol that we show is equivalent to the required spectral gap. In contrast to the model specific computations available previously, the resulting formulas apply to any system in the class without further derivation. A companion implementation evaluates the two normal form coefficients from the coefficient array and verifies the assumptions for a given system. We specialize the formulas to two applications: a strain-gradient regularization of the nonlinear $p$-system, whose quadratic case contains a previously studied model as a special case, and a first order conservative bilinear family, not previously analyzed, in which every bifurcation scenario allowed by the general classification is realized.

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arXiv2018

Admissible Equivalence Transformations for Linearization of Nonlinear Wave Type Equations

In the present paper we consider a general family of two dimensional wave equations which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.

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