اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدKdV hierarchies and quantum Novikovs equations
159
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approach
es. After recalling the derivation of Lowner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarhies (dKP, dBKP, dToda, and dDKP). The one-vaiable reductions are described by solutions of differe
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/el
liptic $N$-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.
This is a short review of the Kadomtsev-Petviashvili hierarchies of types B and C. The main objects are the $L$-operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function and the tau-
function. All of them are discussed in the paper. The connections with the usual Kadomtsev-Petviashvili hierarchy (of the type A) are clarified. Examples of soliton solutions and the dispersionless limit of the hierarchies are also considered.
This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and
negative subgroups L_+, L_- and a commuting sequence in the Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function tau_f for the hierarchy. In this third paper, we prove the following results for the nxn KdV hierarchy: (1) The second partials of ln(tau_f) are differential polynomials of the formal inverse scattering solution u_f. Moreover, u_f can be recovered from the second partials of ln(tau_f). (2) The natural Virasoro action on ln(tau_f) constructed in the second paper is given by partial differential operators in ln(tau_f). (3) There is a bijection between phase spaces of the nxn KdV hierarchy and the Gelfand-Dickey (GD_n) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the nxn KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD_n hierarchy under the bijection.
For each of the simple Lie algebras $mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the all-genera one-point FJRW invariants of $mathfrak{g}$-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic
generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of $mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) case we also find two other normalizations of the sequence that are again integral and of at most exponential growth, and hence conjecturally are the Taylor coefficients of some period functions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
