KdV solves BKP
英文摘要：
In this note, we prove that any tau-function of the KdV hierarchy also solves the BKP hierarchy after a simple rescaling of times.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
Quantum Torus symmetry of the KP, KdV and BKP hierarchies
In this paper, we construct the quantum Torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum Torus Lie algebra in the KP system by acting on its tau function. Comparing to the $W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum Torus symmetries of the KdV and BKP hierarchies and further derive the quantum Torus constraints on their tau functions. These quantum Torus c…
math-phmath.MPnlin.SI
From the $B$-Toda to the BKP hierarchy
It is shown that all $τ$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $τ$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM is a disjoint union of Schubert cells, we classify and give explicit parameterization for points in each Schubert cell by constructing a frame for UOGM in the sense of Sato. $τ$-functions are then expressed in terms of these frames and Schur-Q functions. For concreteness we give a comprehensive study for the $τ$-functions of $B$-Toda which can be viewed as a finit…
math-phmath.COmath.DS
Intersection numbers on $\overline {\mathcal M}_{g,n}$ and BKP hierarchy
In their recent inspiring paper Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar conjecture for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.
hep-thmath-phmath.AG
Gauge transformation and symmetries of the commutative multi-component BKP hierarchy
In this paper, we defined a new multi-component BKP hierarchy which takes values in a commutative subalgebra of $gl(N,\mathbb C)$. After this, we give the gauge transformation of this commutative multi-component BKP (CMBKP) hierarchy. Meanwhile we construct a new constrained CMBKP hierarchy which contains some new integrable systems including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed.
math-phmath.MPnlin.SI
BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions
We derive a formula for the connected $n$-point functions of a tau-function of the BKP hierarchy in terms of its affine coordinates. This is a BKP-analogue of a formula for KP tau-functions proved by Zhou in [arXiv:1507.01679]. Moreover, we prove a simple relation between the KP-affine coordinates of a tau-function $τ(\mathbf{t})$ of the KdV hierarchy and the BKP-affine coordinates of $τ(\mathbf{t}/2)$. As applications, we present a new algorithm to compute the free energies of the Witten-Kontsevich tau-function and the Brézin-Gross-Witten tau-function.
math-phmath.MPnlin.SI
A relationship between rational and multi-soliton solutions of the BKP hierarchy
We consider a special class of solutions of the BKP hierarchy which we call $τ$-functions of hypergeometric type. These are series in Schur $Q$-functions over partitions, with coefficients parameterised by a function of one variable $ξ$, where the quantities $ξ(k)$, $k\in\mathbb{Z^+}$, are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables $ξ(k)$ are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution…
nlin.SI
更多相关论文收起更多论文
$b$-单调Hurwitz数：Virasoro约束、BKP可积层级与$O(N)$-BGW积分
$b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
本文研究单调Hurwitz数的$b$-形变，推导其Virasoro约束并证明相关猜想，在$b=1$时得到Schur展开并验证BKP可积层级，还将模型与$O(N)$版BGW积分关联并显式求解。
math-phmath.COmath.MP
Lattice modified KdV hierarchy from a Lax pair expansion
We produce a hierarchiy of integrable equations by systematically adding terms to the Lax pair for the lattice modified KdV equation. The equations in the hierarchy are related to one aonother by recursion relations. These recursion relations are solved explicitly so that every equation in the hierarchy along with its Lax pair is known.
math-phmath.MPnlin.SI
On the spectral problem of the quantum KdV hierarchy
The spectral problem for the quantum dispersionless Korteweg-de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak-Rossi's quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves …
math-phmath.MPnlin.SI
W型算子在Schur函数和Schur Q函数上的作用
Action of $W$-type operators on Schur functions and Schur Q-functions
本文研究KP和BKP层次对称代数中的W型微分算子，给出其作用于Schur函数和Schur Q函数的统一公式，并据此简化证明Alexandrov猜想与Mironov-Morozov公式。
math-phmath.COmath.MP
Darboux dressing and undressing for the ultradiscrete KdV equation
We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved by obtaining data uniquely characterising the soliton content and the `background' from the initial potential by Darboux transformation.
math-phmath.MPnlin.SI
Cut-and-join description of generalized Brezin-Gross-Witten model
We investigate the Brezin-Gross-Witten model, a tau-function of the KdV hierarchy, and its natural one-parameter deformation, the generalized Brezin-Gross-Witten tau-function. In particular, we derive the Virasoro constraints, which completely specify the partition function. We solve them in terms of the cut-and-join operator. The Virasoro constraints lead to the loop equations, which we solve in terms of the correlation functions. Explicit expressions for the coefficients of the tau-function and the free energy are derived, and a compact formula for the genus zero contribution is conjectured.…
hep-thmath-phmath.CO
Exact accelerating solitons in nonholonomic deformation of the KdV equation with two-fold integrable hierarchy
Recently proposed nonholonomic deformation of the KdV equation is solved through inverse scattering method by constructing AKNS-type Lax pair. Exact and explicit N-soliton solutions are found for the basic field and the deforming function showing an unusual accelerated (decelerated) motion. A two-fold integrable hierarchy is revealed, one with usual higher order dispersion and the other with novel higher nonholonomic deformations.
nlin.SI
Modified KdV hierarchy : Lax pair representation and bi-Hamiltonian structure
We consider equations in the modified KdV (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV equations. We also show that the complex modified KdV (cmKdV) equation follows from the action principle to have a Lagrangian representation. This representation not only provides a basis to write the cmKdV equation in the canonical form endowed with an appropriate Poisson structure but also help us construct a semianalytical solution of it. The solution obtained b…
nlin.SI
