Related papers: Extended Camassa-Holm Hierarchy and Conserved Quan…
This work concerns a study of the quantum mechanical extension of the work of Horwitz et al. [1] on the stability of classical Hamiltonian systems by geometrical methods. Simulations are carried out for several important examples, these…
We describe the physical hypothesis in which an approximate model of water waves is obtained. For an irrotational unidirectional shallow water flow, we derive the Camassa-Holm equation by a variational approach in the Lagrangian formalism.
We study the conservation laws of both the classical and the quantum mechanical continuum $1/r^2$ type systems. For the classical case, we introduce new integrals of motion along the recent ideas of Shastry and Sutherland (SS),…
It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, Ibragimov has shown that…
In this paper, we investigate the long time behaviour for a class of low-regularity solutions of the Camassa-Holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks.
We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of…
A kinetics built upon $q$-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the "quasi-position" whose eigenvalues are the levels of…
In this article we numerically revisit the long-time solution behavior of the Camassa-Holm equation. The finite difference solution of this integrable equation is sought subject to the newly derived initial condition with Delta-function…
In this paper, we review the recently formulated quantum laws of motion and provide new observations. We also extend these laws to higher dimensions. By applying in two dimensions the obtained relations to charge submitted to an electric…
In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an…
The rotation-two-component Camassa--Holm system, which possesses strongly nonlinear coupled terms and high-order differential terms, tends to have continuous nonsmooth solitary wave solutions, such as peakons, stumpons, composite waves and…
In this paper we consider a four-parameter equation including the Camassa-Holm and the Dulling-Gottwald-Holm equations, among others. We prove the existence and uniqueness of solutions to a Cauchy problem involving the equation using Kato's…
The extended flow equations of the multi-component Toda hierarchy are constructed. We give the Hirota bilinear equations and tau function of this new extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms, some extended…
A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second…
The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the $H^1$ and $\dot{H}^1$ right-invariant metrics correspondingly. There is an analogy to…
In this paper, we have constructed the higher order k-bonacci matrices and studied some of their basic properties. We have also shown that these matrices satisfying some new and interesting relations in k-bonacci recurrence. This is the…
Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. These methods are here investigated and…
We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equations content of a Lagrangian theory (in the standard framework of the Calculus of Variations in jet bundles).…
We give conserved quantities of two generalizations of conformal circles on the conformal sphere. One generalization concerns curves carrying a distinguished parallel tractor along them, which can be used to construct the conserved…
A map is presented that associates with each element of a loop group a solution of an equation related by a simple change of coordinates to the Camassa-Holm (CH) Equation. Certain simple automorphisms of the loop group give rise to Backlund…