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Register a new userElliptic Solutions for Higher Order KdV Equations
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We study higher order KdV equations from the GL(2,$mathbb{R}$) $cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.
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We study the spectral (linear) stability and orbital (nonlinear) stability of the elliptic solutions for the focusing modified Korteweg-de Vries (mKdV) equation with respect to subharmonic perturbations and construct the corresponding breather solutions to exhibit the unstable or stable dynamic behavior. The elliptic function solutions of mKdV equation and the fundamental solutions of Lax pair are exactly represented by using the theta function. Based on the `modified squared wavefunction (MSW) method, we construct all linear independent solutions of the linearized KdV equation, and then provide a necessary and sufficient condition of the spectral stability for the elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stable, the orbital stability of the elliptic function solutions with respect to subharmonic perturbations is established under a suitable Hilbert space. Using Darboux-Backlund transformation, we construct the breather solutions to exhibit the unstable or stable dynamic behavior. Through analyzing the asymptotical behavior, we find the breather solution under the $mathrm{cn}$-background is equivalent to the elliptic function solution adding a small perturbation as $ttopminfty$.
We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian $H_k$ of the elliptic Calogero-Moser model which governs the dynamics of poles with respect to the $k$-th hierarchical time. The Hamiltonians $H_k$ are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker-Akhiezer function has essential singularity.
We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians which govern the dynamics of poles with respect to the m-th hierarchical times t_m and bar t_m of the 2D Toda lattice hierarchy are obtained from expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points.
The paper begins with a review of the well known Novikovs equations and corresponding finite KdV hierarchies. For a positive integer $N$ we give an explicit description of the $N$-th Novikovs equation and its first integrals. Its finite KdV hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $mathbb{C}^{2N}$. Then we discuss a non-commutative version of the $N$-th Novikovs equation defined on a finitely generated free associative algebra $mathfrak{B}_N$ with $2N$ generators. In $mathfrak{B}_N$, for $N=1,2,3,4$, we have found two-sided homogeneous ideals $mathfrak{Q}_Nsubsetmathfrak{B}_N$ (quantisation ideals) which are invariant with respect to the $N$-th Novikovs equation and such that the quotient algebra $mathfrak{C}_N = mathfrak{B}_Ndiagup mathfrak{Q}_N$ has a well defined Poincare-Birkhoff-Witt basis. It enables us to define the quantum $N$-th Novikovs equation on the $mathfrak{C}_N$. We have shown that the quantum $N$-th Novikovs equation and its finite hierarchy can be written in the standard Heisenberg form.
Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve equations.
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