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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 description 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.

We introduce the notion of Darboux transformations for the strict KP hierarchy. We previously showed that solutions of this integrable hierarchy can be constructed from a flag variety $$ \mathcal{F} (1)$$ . Here, we describe which two points in this flag variety are connected by such a transformation. Moreover, we present a closed form of the operators that realize this transformation and describe their geometric characteristics. We show which of these Darboux transformations map solutions of the strict $$n$$ -KdV hierarchy to other solutions of this reduction of the strict KP hierarchy.

The nonlocal symmetries are important because they carry information about the existence of Darboux–Backlund and linearizing transformations, and they also allow us to construct nontrivial solutions to systems. In this paper, we focus on investigating three sets of nonlocal symmetries which are realized as appropriate local symmetries of related auxiliary systems for the Korteweg–de Vries hierarchy. Specifically, we construct infinitely many nonlocal symmetries from three different aspects using residues, Lax pairs and quadratic pseudopotentials for each member of the Korteweg–de Vries hierarchy and show how these nonlocal symmetries connect each other. These infinitely many nonlocal symmetries enable us to construct multiple soliton solutions. Moreover, we adapt the multiple soliton solutions derived from nonlocal symmetries and describe a procedure by using velocity resonance mechanism to find molecule solutions, which can be found not only in the optical systems, but also in fluid systems, for the combined third-fifth-order Korteweg–de Vries equation. Several types of molecule structures including soliton molecules, breather molecules and soliton–breather molecules are illustrated.

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak's differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak's equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity., 21 pages, v2: several references and explanations added, published version

Let $$R$$ be a commutative complex algebra and $$ \partial $$ be a $$ \mathbb{C} $$ -linear derivation of $$R$$ such that all powers of $$ \partial $$ are $$R$$ -linearly independent. Let $$R[ \partial ]$$ be the algebra of differential operators in $$ \partial $$ with coefficients in $$R$$ and $$ P{\kern-1.5pt}sd $$ be its extension by the pseudodifferential operators in $$ \partial $$ with coefficients in $$R$$ . In the algebra $$R[ \partial ]$$ , we seek monic differential operators $$ \mathbf{M} _n$$ of order $$n\ge2$$ without a constant term satisfying a system of Lax equations determined by the decomposition of $$ P{\kern-1.5pt}sd $$ into a direct sum of two Lie algebras that lies at the basis of the strict KP hierarchy. Because this set of Lax equations is an analogue for this decomposition of the $$n$$ -KdV hierarchy, we call it the strict $$n$$ -KdV hierarchy. The system has a minimal realization, which allows showing that it has homogeneity properties. Moreover, we show that the system is compatible, i.e., the strict differential parts of the powers of $$M=( \mathbf{M} _n)^{1/n}$$ satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for $$ \mathbf{M} _n$$ and, in particular, for proving that the $$n$$ th root $$M$$ of $$ \mathbf{M} _n$$ is a solution of the strict KP theory if and only if $$ \mathbf{M} _n$$ is a solution of the strict $$n$$ -KdV hierarchy. We characterize the place of solutions of the strict $$n$$ -KdV hierarchy among previously known solutions of the strict KP hierarchy.

The $${\hat{B}}_n^{(1)}$$ -hierarchy is constructed from the standard splitting of the affine Kac–Moody algebra $${\hat{B}}_n^{(1)}$$ , the Drinfeld–Sokolov $${\hat{B}}_n^{(1)}$$ –KdV hierarchy is obtained by pushing down the $${\hat{B}}_n^{(1)}$$ -flows along certain gauge orbit to a cross section of the gauge action. In this paper, we

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered., Comment: It was added a remark after Theorem 1; added Appendix A; added references; two statements moved to Introduction; corrected typos. 26 pages

We use a Hamiltonian normal form approach to study the dynamics of the water wave problem in the small amplitude long wave regime (KdV regime). If $\mu$ is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order $\mu^5$ consists of two decoupled equation, one describing right going waves and the other describing left going waves. Performing a further non Hamiltonian transformation we conjugate each of these equations to a linear combination of the first three equations in the KdV hierarchy. At order $\mu^7$ we find nontrivial terms coupling the two counter-propagating waves., Comment: This second version contains an additional result estimating the error between solutions and an additional result of reduction to KDV5

Recently, we introduced a new class of bounded potentials of the one-dimensional stationary Schrodinger operator on the real axis, and a corresponding family of solutions of the KdV hierarchy. These potentials, which we call primitive, are obtained as limits of rapidly decreasing reflectionless potentials, or multisoliton solutions of KdV. In this note, we introduce generalized primitive potentials, which are obtained as limits of all rapidly decreasing potentials of the Schrodinger operator. These potentials are constructed by solving a contour problem, and are determined by a pair of positive functions on a finite interval and a functional parameter on the real axis.

A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition $$\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _{2m})$$, we show that, for $$k\ge 1$$, $$L_{k}Q_{\lambda } = \sum ^{2m}_{i= 1}(\lambda _i-k)Q_{\lambda -2k\epsilon _i}$$, where $$L_k$$ is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Plucker-like bilinear identity of Q-functions as Pfaffians: $$\sum ^{2m}_{i=2}(-1)^{i}\partial _1Q_{\lambda _1,\lambda _i}\partial _1Q_{\lambda _2, \ldots ,\widehat{\lambda _i},\ldots , \lambda _{2m}}=0$$, where $$\partial _1=\partial /\partial t_1$$. This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.

In this paper, we prove that there are no solutions for the curvature equation \[ \Delta u+e^{u}=8\pi n\delta_{0}\text{ on }E_{\tau}, \quad n\in\mathbb{N}, \] where $E_{\tau}$ is a flat rectangular torus and $\delta_{0}$ is the Dirac measure at the lattice points. This confirms a conjecture in \cite{CLW2} and also improves a result of Eremenko and Gabrielov \cite{EG}. The nonexistence is a delicate problem because the equation always has solutions if $8\pi n$ in the RHS is replaced by $2\pi \rho$ with $0, Comment: 28 pages

The KdV hierarchy is a family of evolutions on a Schr\"odinger operator that preserves its spectrum. Canonical systems are a generalization of Schr\"odinger operators, that nevertheless share many features with Schr\"odinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This suggests that we need a more sophisticated approach to develop such a hierarchy, if it is indeed possible to do so.

In this paper, we show that as $$\tau \rightarrow \sqrt{-1}\infty $$, any zero of the Lame function converges to either $$\infty $$ or a finite point p satisfying $${\text {Re}}p=\frac{1}{2}$$ and $$e^{2\pi i p}$$ being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line $${\text {Re}}z=\frac{1}{2}$$. We also compute explicitly the eigenvalue set of the corresponding $$L^2[0,1]$$ eigenvalue problem for such KdV potentials, thus extends Takemura’s works (Commun Math Phys 235:467–494, 2003) and (Electron J Differ Equ 2004(15):1–30, 2004). Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy et al. (Trans Am Math Soc 358:603–656, 2006) and the Darboux transformation.

In the present contribution, the Hirota-Satsuma coupled KdV hierarchy on noncommutative phase-space is investigated using the noncommutative extension of Lax pair generating technique. In particular, the explicit representation of its associated Lax pair operators is constructed. It is shown that the obtained results for phase-space noncommutativity case reduce back to the standard commutative case when θ goes to ½.

The traveling wave reduction of the Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found. The Lax pair is used to obtain the first integrals of the traveling wave reduction for the Korteweg-de Vries hierarchy. Exact formulas for the first integrals of the hierarchy are given. The first three members of the hierarchy are considered in more detail. These first integrals are the examples of the integrable nonlinear differential equations with the Painleve property. Exact solutions in the form of the soliton for the traveling wave reduction of the Korteweg-de Vries hierarchy and its first integrals are presented.

We establish precise spectral criteria for potential functions $V$ of reflectionless Schr\"odinger operators $L_V = -\partial_x^2 + V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition., Comment: 52 pages, 1 figure

A self-similar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented. This Lax pair can be used to solve the Cauchy problem for the studied hierarchy. It is shown that special solutions of the self-similar reduction of the KdV hierarchy are determined via the transcendents of the first Painleve hierarchy. Rational solutions are expressed by means of the Yablonskii–Vorob’ev polynomials. The map of the solutions for the second Painleve hierarchy into the solutions for the self-similar reduction of the KdV hierarchy is illustrated using the Miura transformation. Lax pairs for equations of the hierarchy for the Yablonskii–Vorob’ev polynomial are discussed. Special solutions to the hierarchy for the Yablonskii–Vorob’ev polynomials are given. Some other hierarchies with properties of the Painleve hierarchies are presented. The list of nonlinear differential equations whose general solutions are expressed in terms of nonclassical functions is extended.

Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particular we prove that this is indeed the case for \begin{document}$ 2 $\end{document} -soliton solutions of the fourth-order stationary equation.

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 of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group., Comment: 21 pages

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten--Kontsevich tau-function, to the generalized Br\'ezin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lam\'e tau-function are also given., Comment: Minor changes: added remarks, added references, corrected typos; 32 pages

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathscr{H}$ manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post-Winternitz system and of a stationary flow of the KdV hierarchy., Comment: 32 pages

We study the duality between JT gravity and the double-scaled matrix model including their respective deformations. For these deformed theories we relate the thermal partition function to the generating function of topological gravity correlators that are determined as solutions to the KdV hierarchy. We specialise to those deformations of JT gravity coupled to a gas of defects, which conforms with known results in the literature. We express the (asymptotic) thermal partition functions in a low temperature limit, in which non-perturbative corrections are suppressed and the thermal partition function becomes exact. In this limit we demonstrate that there is a Hawking-Page phase transition between connected and disconnected surfaces for this instance of JT gravity with a transition temperature affected by the presence of defects. Furthermore, the calculated spectral form factors show the qualitative behaviour expected for a Hawking-Page phase transition. The considered deformations cause the ramp to be shifted along the real time axis. Finally, we comment on recent results related to conical Weil-Petersson volumes and the analytic continuation to two-dimensional de Sitter space., Comment: 42 pages, 4 figures; v2: minor modifications

We study thermal correlation functions of Jackiw-Teitelboim (JT) supergravity. We focus on the case of JT supergravity on orientable surfaces without time-reversal symmetry. As shown by Stanford and Witten recently, the path integral amounts to the computation of the volume of the moduli space of super Riemann surfaces, which is characterized by the Brezin-Gross-Witten (BGW) tau-function of the KdV hierarchy. We find that the matrix model of JT supergravity is a special case of the BGW model with infinite number of couplings turned on in a specific way, by analogy with the relation between bosonic JT gravity and the Kontsevich-Witten (KW) model. We compute the genus expansion of the one-point function of JT supergravity and study its low-temperature behavior. In particular, we propose a non-perturbative completion of the one-point function in the Bessel case where all couplings in the BGW model are set to zero. We also investigate the free energy and correlators when the Ramond-Ramond flux is large. We find that by defining a suitable basis higher genus free energies are written exactly in the same form as those of the KW model, up to the constant terms coming from the volume of the unitary group. This implies that the constitutive relation of the KW model is universal to the tau-function of the KdV hierarchy., 36 pages, v2: typos corrected, footnotes and references added, v3: several explanations added, typos corrected, published version

We present the hierarchy and soliton solutions associated with a multi-component generalisation of the modified Korteweg–de Vries equation. A recursive formula for obtaining the Lax operators associated with the higher flows of the hierarchy is provided. Using the method of rational dressing and the symmetries of the Lax operators we obtain the one-soliton solution. We also derive the general rank one-breather solution and express it in terms of determinants. We present the simplest case of the one-breather solution, which is parameterised by two constant unit vectors normal to each other. Finally, we obtain the associated Backlund transformation for the hierarchy.