Special Issue in Memory of Decio Levi

Edited by Giorgio Gubbiotti, Orlando Ragnisco, Paolo Maria Santini and Federico Zullo

1. High order multiscale analysis of discrete integrable equations

Rafael Hernandez Heredero ; Decio Levi ; Christian Scimiterna.
In this article we present the results obtained applying the multiple scale expansion up to the order $\varepsilon^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the integrability conditions given by the multiple scale expansion give rise to 4 nonlinear equations, 3 of which seem to be new, depending at most on 2 parameters.

2. Conjectures analogous to the Collatz conjecture

Fabio Briscese ; Francesco Calogero.
In this paper we introduce some conjectures analogous to the well-known Collatz conjecture.

3. Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

E. I. Kaptsov ; V. A. Dorodnitsyn.
Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

4. Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

Paz Albares ; Pilar G. Estévez ; Alejandro González-Parra ; Paula del Olmo.
This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $\tau$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

5. On k-wave solutions of quasilinear systems of partial differential equations

Alfred Michel Grundland.
In this paper, we establish a relation between two seemingly unrelated concepts for solving first-order hyperbolic quasilinear systems of partial differential equations in many dimensions. These concepts are based on a variant of the conditional symmetry method and on the generalized method of characteristics. We present the outline of recent results on multiple Riemann wave solutions of these systems. An auxiliary result concerning a modification of the Frobenius theorem for integration is used. We apply this result in order to show that the conditional symmetry method can deliver larger classes of multiple Riemann wave solutions, through a simpler procedure, than the one obtained from the generalized method of characteristics. We demonstrate that solutions can be interpreted physically as a superposition of k single waves. These theoretical considerations are illustrated by examples of hydrodynamic-type systems in (n+1) dimensions.

6. A new approach to integrals of discretizations by polarization

Yuri B. Suris.
Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.

7. Bäcklund transformations as integrable discretization. The geometric approach

Adam Doliwa.
We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by Bäcklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that Bäcklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.

8. Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation

Edoardo Peroni ; Jing Ping Wang.
In this paper we study the algebraic properties of a new integrable differential-difference equation. This equation can be seen as a deformation of the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt equation in its continuous limit. Using its Lax representation we explicitly construct a recursion operator for this equation and prove that it is a Nijenhuis operator. Moreover, we present the bi-Hamiltonian structures for this new equation.

9. Negative flows and non-autonomous reductions of the Volterra lattice

V. E. Adler.
We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as $(m+1)$-component difference equations of the Painlevé type generalizing the dP$_1$ and dP$_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the Bäcklund transformations which form the $\mathbb{Z}^m$ lattice.

10. KdV hierarchies and quantum Novikov's equations

V. M. Buchstaber ; A. V. Mikhailov.
This paper begins with a review of the well-known KdV hierarchy, the $N$-th Novikov equation, and its finite hierarchy in the classical commutative case. This finite hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $\mathbb{C}^{2N}$. We discuss a non-commutative version of the $N$-th Novikov hierarchy defined on the finitely generated free associative algebra ${\mathfrak{B}}_N$ with $2N$ generators. Using the method of quantisation ideals in ${\mathfrak{B}}_N$, for $N=1,2,3,4$, we obtain two-sided homogeneous ideals ${\mathfrak{Q}}_N\subset{\mathfrak{B}}_N$ (quantisation ideals) that are invariant with respect to the $N$-th Novikov equation and such that the quotient algebra ${\mathfrak{C}}_N = {\mathfrak{B}}_N/ {\mathfrak{Q}}_N$ has a well-defined Poincare-Birkhoff-Witt basis. This allows us to define the quantum $N$-th Novikov equation and its hierarchy on ${\mathfrak{C}}_N$. We derive $N$ commuting quantum first integrals (Hamiltonians) and represent the equations of the hierarchy in the Heisenberg form. Essential for our research is the concept of cyclic Frobenius algebras, which we introduced in our recent paper. In terms of the quadratic form that defines the structure of a cyclic Frobenius algebra, we explicitly express the first integrals of the $N$-th Novikov hierarchy in the commutative, free, and quantum cases.

11. Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

Xingbiao Hu ; Guofu Yu ; Yingnan Zhang.
In this paper, we give a procedure for discretizing recursion operators by utilizing unified bilinear forms within integrable hierarchies. To illustrate this approach, we present unified bilinear forms for both the AKNS hierarchy and the KdV hierarchy, derived from their respective recursion operators. Leveraging the inherent connection between soliton equations and their auto-Bäcklund transformations, we discretize the bilinear integrable hierarchies and derive discrete recursion operators. These discrete recursion operators exhibit convergence towards the original continuous forms when subjected to a standard limiting process.

12. An exercise in experimental mathematics: calculation of the algebraic entropy of a map

C. M. Viallet.
We illustrate the use of the notion of derived recurrences introduced earlier to evaluate the algebraic entropy of self-maps of projective spaces. We in particular give an example, where a complete proof is still awaited, but where different approaches are in such perfect agreement that we can trust we get to an exact result. This is an instructive example of experimental mathematics.

13. Degree growth of lattice equations defined on a 3x3 stencil

Jarmo Hietarinta.
We study complexity in terms of degree growth of one-component lattice equations defined on a $3\times 3$ stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type. Initial values are given on a staircase or on a corner configuration and depend linearly or rationally on a special variable, for example $f_{n,m}=\alpha_{n,m}z+\beta_{n,m}$, in which case we count the degree in $z$ of the iterates. Known integrable cases have linear growth if only one initial values contains $z$, and quadratic growth if all initial values contain $z$. Even a small deformation of an integrable equation changes the degree growth from polynomial to exponential, because the deformation will change factorization properties and thereby prevent cancellations.

14. A new discretization of the Euler equation via the finite operator theory

Miguel A. Rodríguez ; Piergiulio Tempesta.
We propose a novel discretization procedure for the classical Euler equation, based on the theory of Galois differential algebras and the finite operator calculus developed by G.C. Rota and collaborators. This procedure allows us to define algorithmically a new discrete model which inherits from the continuous Euler equation a class of exact solutions.

15. Soliton equations: admitted solutions and invariances via Bäcklund transformations

Sandra Carillo ; Cornelia Schiebold.
A couple of applications of Bäcklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on proving a new invariance admitted by a third order nonlinear evolution equation and, on the other one, on the construction of solutions. Indeed, via Bäcklund transformations, a {\it Bäcklund chart}, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide Bäcklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the {\it Korteweg-deVries interacting soliton} (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian {\it Bäcklund chart}, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.

16. On the Lagrangian multiform structure of the extended lattice Boussinesq system

F. W. Nijhoff ; D. J. Zhang.
The lattice Boussinesq (lBSQ) equation is a member of the lattice Gel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which is an infinite family of integrable systems of partial difference equations labelled by an integer $N$, where $N=2$ represents the lattice Korteweg-de Vries (KdV) system, and $N=3$ the Boussinesq system. In \cite{Hiet2011} it was shown that, written as three-component system, the lBSQ system allows for extra parameters which essentially amounts to building the lattice KdV inside the lBSQ. In this paper we show that, on the level of the Lagrangian structure, this boils down to a linear combination of Lagrangians from the members of the lGD hierarchy as was established in \cite{LobbNijGD2010}. The corresponding Lagrangian multiform structure is shown to exhibit a `double zero' structure.

17. Algebraic entropy for systems of quad equations

Giorgio Gubbiotti.
In this work I discuss briefly the calculation of the algebraic entropy for systems of quad equations. In particular, I observe that since systems of multilinear equations can have algebraic solution, in some cases one might need to restrict the direction of evolution only to the pair of vertices yielding a birational evolution. Some examples from the exiting literature are presented and discussed within this framework.

18. Integrable maps in 4D and modified Volterra lattices

A. N. W. Hone ; J. A. G. Roberts ; P. Vanhaecke ; F. Zullo.
In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus $g$ algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted ${\cal V}_g$. The original motivation behind our work was the fact that, in the particular case $g=2$, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of […]

19. An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

Giorgio Gubbiotti ; David McLaren ; G. R. W. Quispel.
We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.