## ] o c n m p [## Open Communications in Nonlinear Mathematical Physics |

In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider Stäckel transform between two pairs of Stäckel systems, generated by 2n-parameter algebraic curves on the plane, as well as Miura maps between Stäckel systems generated by (n+N)-parameter algebraic curves, leading to multi-Hamiltonian representation of these systems.

We present a construction of a class of rational solutions of the Painlevé V equation that exhibit a two-fold degeneracy, meaning that there exist two distinct solutions that share identical parameters. The fundamental object of our study is the orbit of translation operators of $A^{(1)}_{3}$ affine Weyl group acting on the underlying seed solution that only allows action of some symmetry operations. By linking points on this orbit to rational solutions, we establish conditions for such degeneracy to occur after involving in the construction additional Bäcklund transformations that are inexpressible as translation operators. This approach enables us to derive explicit expressions for these degenerate solutions. An advantage of this formalism is that it easily allows generalization to higher Painlevé systems associated with dressing chains of even period $N>4$.

We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are integrable through spectral methods, typically belonging to the QRT family, to a variety of linearisable ones. We show that by judiciously choosing these linearisable mappings, it is possible to obtain higher order mappings that exhibit the maximal degree growth compatible with integrability, i.e. for which the degree grows as a polynomial of order equal to the order of the mapping. In all the cases we analysed, we found that maximal degree growth was associated with the existence of an unconfining singularity pattern. Several cases with submaximal growth but which still possess unconfining singularity patterns are also presented. In many cases the exact degrees of the iterates of the mappings were obtained by applying a method due to Halburd, based on the preimages of specific values that appear in the singularity patterns of the mapping, but we also present some examples where such a calculation appears to be impossible.

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat {\cal{G}} $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping […]

A seventh order ordinary differential equation (ODE) arising by reduction of the Drinfeld-Sokolov hierarchyis shown to be identical to a similarity reduction of an equationin the hierarchy of Sawada-Kotera.We also exhibit its link with a particular F-VI,a fourth order ODE isolated by Cosgrove which is likely to define a higher order Painlevé function.

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.

This work is an analytical investigation of the evolution of surface water waves in Miles and Jeffreys theories of wind wave interaction in water of finite depth. The present review is divided into two major parts. The first corresponds to the surface water waves in a linear regime and its nonlinear extensions. In this part, Miles theory of wave amplification by wind is extended to the case of finite depth. The dispersion relation provides a wave growth rate depending on depth. Our theoretical results are in good agreement with the data from the Australian Shallow Water Experiment and the data from the Lake George experiment. In the second part of this study, Jeffreys theory of wave amplification by wind is extended to the case of finite depth, where the Serre-Green-Naghdi is derived. We find the solitary wave solution of the system, with an increasing amplitude under the action of the wind. This continuous increase in amplitude leads to the soliton breaking and blow-up of the surface wave in finite time. The theoretical blow-up time is calculated based on actual experimental data. By applying an appropriate perturbation method, the SGN equation yields Korteweg de Vries Burger equation (KdVB). We show that the continuous transfer of energy from wind to water results in the growth of the KdVB soliton amplitude, velocity, acceleration, and energy over time while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys approach due to […]

We derive the general conditions for fully-nonlinear symmetry-integrable second-order evolution equations and their first-order recursion operators. We then apply the established Propositions to find links between a class of fully-nonlinear third-order symmetry-integrable evolution equations and fully-nonlinear second-order symmetry-integrable evolution equations.

We construct soliton solutions of the four-dimensional Wess-Zumino-Witten (4dWZW) model in the context of a unified theory of integrable systems with relation to the 4d/6d Chern-Simons theory. We calculate the action density of the solutions and find that the soliton solutions behave as the KP-type solitons, that is, the one-soliton solution has a localized action/energy density on a 3d hyperplane in 4-dimensions (soliton wall) and the n-soliton solution describes n intersecting soliton walls with phase shifts. We note that the Ward conjecture holds mostly in the split signature (+,+,-,-). Furthermore, the 4dWZW model describes the string field theory action of the open N=2 string theory in the four-dimensional space-time with the split signature and hence our soliton solutions would describe a new-type of physical objects in the N=2 string theory. We discuss instanton solutions in the 4dWZW model as well. Noncommutative extension and quantization of the unified theory of integrable systems are also discussed.

The Kontsevich star-product admits a well-defined restriction to the class of affine -- in particular, linear -- Poisson brackets; its graph expansion consists only of Kontsevich's graphs with in-degree $\leqslant 1$ for aerial vertices. We obtain the formula $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ with harmonic propagators for the graph weights (over $n\leqslant 7$ aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet--Felder--Willwacher, that they match the computations using the $\textsf{kontsevint}$ software by Panzer, and the resulting affine star-product is associative modulo $\bar{o}(\hbar^7)$. We discover that the Riemann zeta value $\zeta(3)^2/\pi^6$, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ \textit{because} all the $\mathbb{Q}$-linear combinations of Kontsevich graphs near $\zeta(3)^2/\pi^6$ represent differential consequences of the Jacobi identity for the affine Poisson bracket, hence their contribution vanishes. We thus derive a ready-to-use shorter formula $\star_{\text{aff}}^{\text{red}}$ mod~$\bar{o}(\hbar^7)$ with only rational coefficients.