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Publication# A class of Neumann type systems and its application

Abstract

A class of Neumann type systems are derived separating the spatial and temporal variables for the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation and the modified Korteweg-de Vries (mKdV) hierarchy. The Lax-Moser matrix of Neumann type systems is worked out, which generates a sequence of integrals of motion and a hyperelliptic curve of KdV type. We deduce the constrained Hamiltonians to put Neumann type systems into canonical Hamiltonian equations and further complete the Liouville integrability for the Neumann type systems. We also specify the relationship between Neumann type systems and infinite dimensional integrable systems (IDISs), where the involutivity solutions of Neumann type systems yield the finite parametric solutions of IDISs. From the Abel-Jacobi variables, the evolution behavior of Neumann type flows are shown on the Jacobian of a Riemann surface. Finally, the Neumann type flows are applied to produce some explicit solutions expressed by Riemann theta functions for the 2+1 dimensional CDGKS equation and the mKdV hierarchy.

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In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages

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Florian Paul Robert Maurin, Alessandro Spadoni

Wave propagation in pinned-supported, post-buckled beams can be described with the Korteweg de Vries (KdV) equation. Finite-element simulations however show that the KdV is applicable only to post-buckled beams with strong pre-compression. For weak and moderate pre-stress, a dispersive front is present and it is the aim of the current paper to analyze sources of dispersion beyond periodicity given three support types: guided, pinned, and free. Bloch theorem and a transfer-matrix method are employed to obtain numerical dispersion relations and characteristic wave modes, which are used to analyze the effects of pre-stress, initial curvature, and the influence of support types. Additionally, a new method is proposed to obtain a semi-analytical dispersion equation for the acoustic branch. Powers of frequency and the propagation constant are explicitly expressed and their coefficients are based on stiffness and mass-matrix components obtained from finite elements. This allows a physical interpretation of the dispersion sources, based on which, equivalent mass-spring models of post-buckled beam are proposed. It is found that mass and stiffness coupling are significant dispersion sources. In the present paper, a reduced form of Bloch theorem is presented exploiting glide-reflection symmetries, reducing the size of the unit cell and allowing an easier representation and interpretation of results.

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2021