On applications of generalized Bäcklund transformations to continuum mechanics

On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections

“…It would accordingly be of interest to investigate the application of the present class of reciprocal transformations of two-dimensional, steady relativistic gasdynamics with such model pressure-energy density relations. The fact that the original gasdynamic reciprocal transformations of Bateman [2] were shown in [3] to constitute particular Bäcklund transformations [5,6] suggests as a potential consequence of the present work that the wide class of Bäcklund transformations of the type systematically applied by Loewner to the hodograph equations of classical gasdynamics may be applied, in turn, to the hodograph system of two-dimensional relativistic gasdynamics as set down in [16]. This with the view to the reduction to tractable form of this system for appropriate model constitutive laws.…”

“…Bateman [4] later noted that both the adjoint and reciprocal transformations constitute particular Bäcklund transformations (e.g. [5,6]). Loewner [7,8], in subsequent work, investigated the application of a broad class of matrix Bäcklund transformations which allow the reduction of the classical hodograph system of irrotational plane gasdynamics to tractable canonical form in subsonic, transonic or supersonic régimes.…”

Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

“…Reciprocal transformations are transformations of independent variables of evolutionary PDEs, which originally appear in the study of gas dynamics, see [40] and references therein. Such transformations also play important roles in the study of the theory of integrable hierarchies.…”

Bäcklund transformations between four Lax-integrable 3D equations

“…The purpose of this paper is to construct a Lax representation for rmhd. Lax representations provide the basic construction that allows applications of a number of techniques for studying nonlinear partial differential equations, whence they are considered as the key feature indicating integrability thereof, see [30,32,24,22,7,1,20,23,2] and references therein. From the viewpoint of geometry of differential equations, Lax representations and other nonlocal structures of integrable nonlinear systems are naturally formulated in the language of differential coverings, [9,10,29].…”