The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.
In this paper, we develop the mathematical framework to describe the physical phenomenon behind the equilibrium configuration joining two antiferromagnetic domains. We firstly define the total energy of the system and deduce the governing equations by minimizing it with respect to the field variables. Then, we solve the resulting system of nonlinear PDEs together with proper initial and boundary conditions by varying the orientation of the 90 • domain wall (DW) configuration along the sample. Finally, the angular dependence of elastic and magnetoelastic energies as well as of incompatibilitydriven volume effects is computed.
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