KdV Hierarchy Research Articles

On the Generalized KdV Hierarchy and Boussinesq Hierarchy with Lax Triple

Based on the Nambu 3-bracket and the operators of the KP hierarchy, we propose the generalized Lax equation of the Lax triple. Under the operator constraints, we construct the generalized KdV hierarchy and Boussinesq hierarchy. Moreover, we present the exact solutions of some nonlinear evolution equations.

Restrictions on the Existence of a Canonical System Flow Hierarchy

The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheless share many features with Schrödinger 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.

Haantjes algebras of classical integrable systems

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.

Extended r-spin theory in all genera and the discrete KdV hierarchy

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the r-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus 0 it corresponds to a flat F-manifold. For r=2 we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.

On determining some exact wave solutions to the Nizhnik–Novikov–Veselov system via a rebuts technique

In this study, a significant model of mathematical physics used to describe real-world problems, namely the Nizhnik–Novikov–Veselov (NNV) system is studied and various versions of wave solutions are simultaneously explored. The NNV system is considered an essential model to the KdV hierarchy and is widely used to study the scalar nucleons associated with neutral scalar masons. Thus, the exact wave solutions are important to the associated researchers. To this aim, a robust approach named the generalized exponential rational function method (GERFM) is used to achieve this goal. With the aid of GERFM abundant new wave solutions are formally constructed. All generated solutions are constructed with distinct forms that are dependent on free parameters and determined by the self-form of the NNV system. Moreover, the free parameters play the main roles in the dispersive terms and the values of free parameters influence the physical properties of the traveling solitary and periodic waves to the NNV system directly. Finally, some 3D graphs of the generated solutions are provided to show the propagations of traveling waves. The performed results will have great potential applications to optical fibers, magneto-electrodynamics, the collision of heavy ions, and quantum mechanics.

Nonlocal symmetries and molecule structures of the KdV 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.

Space Curves and Solitons of the KP Hierarchy. I. The l-th Generalized KdV Hierarchy

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\dots, lm+k\rangle$, where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.

Genus expansion of open free energy in 2d topological gravity

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.

On tau-functions for the KdV hierarchy

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 Brézin–Gross–Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lamé tau-function are also given.

Quadratic double ramification integrals and the noncommutative KdV hierarchy

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp(\mu^2 \Theta)$ where $\mu$ is a parameter and $\Theta$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank $1$ integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.

Reductions of the strict KP hierarchy

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.

Virasoro symmetries of multicomponent Gelfand–Dickey systems

In this paper, we mainly study the additional symmetry and $\tau$ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems, such as the multi-component KdV hierarchy and the multi-component Boussinesq hierarchy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our research, the additional flows of the additional symmetries can not all survive. By calculating, we find that the generator of the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same.

JT supergravity and Brezin-Gross-Witten tau-function

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.

Hodge–GUE Correspondence and the Discrete KdV Equation

We prove the conjectural relationship recently proposed in Dubrovin and Yang (Commun Number Theory Phys 11:311–336, 2017) between certain special cubic Hodge integrals of the Gopakumar–Marino–Vafa type (Gopakumar and Vafa in Adv Theor Math Phys 5:1415–1443, 1999, Marino and Vafa in Contemp Math 310:185–204, 2002) and GUE correlators, and the conjecture proposed in Dubrovin et al. (Adv Math 293:382–435, 2016) that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy.

Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows

We consider the Cauchy problem for the KdV hierarchy – a family of integrable PDEs with a Lax pair representation involving one-dimensional Schrödinger operators – under a local in time boundedness assumption on the solution. For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency. This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craig-type conditions.

Darboux Transforms for the $${\hat{B}}_n^{(1)}$$-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

Liouville correspondences between multicomponent integrable hierarchies

In this paper, we establish Liouville correspondences for the integrable two-component Camassa-Holm hierarchy, the two-component Novikov (Geng-Xue) hierarchy, and the two-component dual dispersive water wave hierarchy by means of the related Liouville transformations. This extends previous results on the scalar Camassa-Holm and KdV hierarchies, and the Novikov and Sawada-Kotera hierarchies to the multi-component case.

Matrix Resolvent and the Discrete KdV Hierarchy

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.

Factorization of KdV Schrödinger operators using differential subresultants

We address the classical factorization problem of a one dimensional Schrödinger operator −∂2+u−λ, for a stationary potential u of the KdV hierarchy but, in this occasion, a “parameter” λ is considered. Inspired by the more effective approach of Gesztesy and Holden to the “direct” spectral problem in [14], we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization, that in addition allows one parameter form factorizations by means of suitable global parametrizations of the spectral curve. Differential resultants are used for computing spectral curves, and differential subresultants to obtain the first order common factor. To make our method fully effective, we design a symbolic algorithm to compute the integration constants of the KdV hierarchy, in the case of KdV potentials that become rational under a Hamiltonian change of variable. Explicit computations are carried for Schrödinger operators with solitonic potentials.

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.