Table of contents

Modern theory of nonlinear integrable equations is nowdays an important and effective tool of study for numerous nonlinear phenomena in various branches of physics from hydrodynamics and optics to quantum filed theory and gravity. It includes the study of nonlinear partial differential and discrete equations, regular and singular behaviour of their solutions, Hamitonian and bi- Hamitonian structures, their symmetries, associated deformations of algebraic and geometrical structures with applications to various models in physics and mathematics.

The PMNP 2013 conference focused on recent advances and developments in

• Continuous and discrete, classical and quantum integrable systems

• Hamiltonian, critical and geometric structures of nonlinear integrable equations

• Integrable systems in quantum field theory and matrix models

• Models of nonlinear phenomena in physics

• Applications of nonlinear integrable systems in physics

The Scientific Committee of the conference was formed by

• Francesco Calogero (University of Rome `La Sapienza', Italy)

• Boris A Dubrovin (SISSA, Italy)

• Yuji Kodama (Ohio State University, USA)

• Franco Magri (University of Milan `Bicocca', Italy)

• Vladimir E Zakharov (University of Arizona, USA, and Landau Institute for Theoretical Physics, Russia)

The Organizing Committee: Boris G Konopelchenko, Giulio Landolfi, Luigi Martina, Department of Mathematics and Physics `E De Giorgi' and the Istituto Nazionale di Fisica Nucleare, and Raffaele Vitolo, Department of Mathematics and Physics `E De Giorgi'.

A list of sponsors, speakers, talks, participants and the conference photograph are given in the PDF.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schrödinger–like equation which provides a semiclassical Hamiltonian picture of the evolution of a 'quantum of space', as shown by the authors in [1]. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuous) variable. Key tool for this comparative analysis is the duality operation defined on the generators of the quadratic algebra and suitably extended to the various families of overlap functions (generalized recoupling coefficients). These families, recognized as lying at the top level of the Askey scheme, are classified and a few limiting cases are addressed.

We present examples of Lax-integrable multi-dimensional systems of partial differential equations with higher local symmetries. We also consider Lagrangian deformations of these equations and construct variational bivectors on them.

In this paper we describe some recent progresses in the study of the leading quantum correction at strong coupling of the dressing phase appearing in the Bethe Ansatz for string theory on the AdS₃ × S³ × T⁴ background. Using the SU(2) rigid circular string as guiding example, we find that the phase is different than in the AdS_5,4 cases. We discuss in detail the determination of the phase using both a Word-Sheet approach and the Algebraic Curve formalism.

We consider the integrable system of isotropic SU(3) Landau-Lifshits equation as a Hamiltonian system on a coadjoint orbit of the SU(3) loop group. We connect the mentioned equation with an isotropic SU(3) magnet because it describes the mean fields of magnetic and quadrupole moments in a spin-1 lattice. For the system of isotropic SU(3) Landau-Lifshits equation we perform separation of variables in Sklyanin's manner, and integrate in the lowest finite gap where the spectral curve is elliptic.

We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential m-form Ω_m with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations.

With the aim of presenting a unified viewpoint for the variational and Hamiltonian formalism of two-dimensional incompressible stratified Euler equations, we revisit some of the formulations currently discussed in the literature and examine their mutual relations. We concentrate on the example of two-layered systems and its one-dimensional reduction, and use it to illustrate general consequences of density stratification on conservation laws which have been partially overlooked until now. In particular, we focus on the conservation of horizontal momentum for stratified ideal fluid motion under gravity between rigid lids.

Nonlinear interactions among small amplitude, long wavelength, obliquely propagating waves on the surface of shallow water often generate web-like patterns. In this article, we discuss how line-soliton solutions of the Kadomtsev-Petviashvili (KP) equation can approximate such web-pattern in shallow water wave. We describe an "inverse problem" which maps a certain set of measurable data from the solitary waves in the given pattern to the parameters required to construct an exact KP soliton that describes the non-stationary dynamics of the pattern. We illustrate the inverse problem using explicit examples of shallow water wave pattern.

We present here a rigorous proof of the quasi steady state approximation in the context of a system of reaction-diffusion equations coming out of a typical chain of irreversible chemical reactions.

We derive a class of equations of state for a multi-phase thermodynamic system associated with a finite set of order parameters that satisfy an integrable system of hydrodynamic type. As particular examples, we discuss one-phase systems such as the van der Waals gas and the effective molecular field model. The case of N–phase systems is also discussed in detail in connection with entropies depending on the order parameter according to Tsallis' composition rule.

Via a "tropical limit" (Maslov dequantization), Korteweg-deVries (KdV) solitons correspond to piecewise linear graphs in two-dimensional space-time. We explore this limit.

General scheme for calculations via Zakharov and Manakov -dressing method of exact solutions, nonstationary and stationary, of Nizhnik-Veselov-Novikov (NVN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u⁽ⁿ⁾, n = 1,..., N is presented. Simple nonlinear superpositions are given up to a constant by the sums of solutions u⁽ⁿ⁾ and calculated by -dressing of the first auxiliary linear problem with nonzero asymptotic values of potential at infinity. It is remarkable that in the zero limit of asymptotic values of potential simple nonlinear superpositions convert in to linear ones in the form of the sums of special solutions u⁽ⁿ⁾. It is shown that the sums u = u^(k_l) + ...+ u^(k_m), 1 ≤ k₁ < k₂ < ... < k_m ≤ N of arbitrary subsets of these N solutions are also exact solutions of NVN equation. The obtained results are illustrated in detail by hyperbolic version of NVN equation, i. e. by NVN-II equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions.

The nonlinear problem of the wave propagation is considered. In addition to Kerr nonlinearity the question of the existence of concentrated solutions is analyzed for the threshold and saturable nonlinearity. It is shown that both in the case of threshold nonlinearity, and in the case of saturable nonlinearity solitary waves – concentrated solutions of the corresponding wave equations exist. For the nonlocal nonlinearity, it is taken into account that the diffusion process transforms the interaction of the electromagnetic field with the environment. This phenomenon is described by the system of differential equations including the equation for the perturbation of the dielectric permittivity. The mathematical problem is reduced to the eigenvalue problem for nonlinear integro-differential equation of Hartree type. The computational procedure is constructed.

The self-modulation, resulting from its interaction with the surrounding medium, of a relativistic charged-particle beam traveling through an overdense plasma, is investigated theoretically. The description of the transverse nonlinear and collective beam dynamics of an electron (or positron) beam in a plasma-based accelerator is provided in terms of a thermal matter wave envelope propagation. This is done using the quantum-like description provided by the thermal wave model. It is shown that the charged-particle beam dynamics is governed by a Zakharov-type system of equations, comprising a nonlinear Schrödinger equation that is governing the spatiotemporal evolution of the thermal matter wave envelope and a Poisson-like equation for the wake potential that is generated by the bunch itself.

The most challenging problem in the implementation of the so-called unified transform to the analysis of the nonlinear Schrödinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called linearizable boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large t, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large t behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given τ-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular τ-periodic functions, which includes {a exp(iωt)|a > 0, ω ≥ a²}, then the large t behavior of the Neumann value is given by a τ-periodic function which can be computed explicitly.

We study the solutions of the one dimensional focusing NLS equation. Here we construct new deformations of the Peregrine breather of order 7 with 12 real parameters. We obtain new families of quasi-rational solutions of the NLS equation. With this method, we construct new patterns of different types of rogue waves. We recover triangular configurations as well as rings isolated. As already seen in the previous studies, one sees appearing for certain values of the parameters, new configurations of concentric rings.

We use Riemann-Hilbert Problems with canonical normalization to develop technique for constructing families of commuting operators. As a result we are able to derive new hierarchies of integrable nonlinear evolution equations.

The G-strand equations for a map × into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t, s) : × → G, where t and s are the independent variables of the G-strand equations. The Euler-Poincaré reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra and its co-algebra, * with respect to the pairing provided by the variational derivatives of the Lagrangian.

We review examples of different G-strand constructions, including matrix Lie groups and diffeomorphism group. In some cases the G-strand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.

Deterministic dynamical system which has an asymptotical stable equilibrium is considered under persistent perturbation by white noise. It is well known that if the perturbation does not vanish in the equilibrium position then there is not Lyapunov's stability. The trajectories of the perturbed system diverge from the equilibrium to arbitrarily large distances with probability 1 in finite time. New concept of stability on a large time interval is discussed. The length of interval agrees the reciprocal quantity of the perturbation parameter. The measure of stability is the expectation of the square distance from the trajectory till the equilibrium position. The method of parabolic equation is applied to both estimate the expectation and prove such stability. The main breakthrough is the barrier function derived for the parabolic equation. The barrier is constructed by using the Lyapunov function of the unperturbed system.

Cubic nonlinear Schrödinger type equation with specific initial-boundary conditions in the infinite domain is considered. The equation is reduced to an equivalent system of partial differential equations and studied in the case of solitary waves. The system is modified by introducing new functions, one of which belongs to the class of functions of negligible fifth order and vanishing at infinity exponentially. For this class of functions the system is reduced to a nonlinear elliptic equation which can be solved analytically, thereby allowing us to present nontrivial approximated solutions of nonlinear Schrödinger equation. These solutions describe a new class of symmetric solitary waves. Graphics of modulus of the corresponding wave function are constructed by using Maple.

We re-address the problem of construction of new infinite-dimensional completely integrable systems on the basis of known ones, and we reveal a working mechanism for such transitions. By splitting the problem's solution in two steps, we explain how the classical technique of Gardner's deformations facilitates – in a regular way – making the first, nontrivial move, in the course of which the drafts of new systems are created (often, of hydrodynamic type). The other step then amounts to higher differential order extensions of symbols in the intermediate hierarchies (e. g., by using the techniques of Dubrovin et al. [1, 2] and Ferapontov et al. [3, 4]).

Focusing on the problem of finding the right number of infinite series of conserved charges in integrable models with infinite degrees of freedom, we explore the well known nonlinear Scrödinger (NLS) equation discovering novel sets of charges with hidden integrable hierarchies. This prompts us to construct a new integrable NLS equation in 2 + 1-dimensions. Few important applications of our result including a 2D analytic model for the ocean rogue wave are reported.

Analytical form of quantum corrections to quasi-periodic solution of Sine-Gordon model and periodic solution of ϕ⁴ model is obtained through zeta function regularisation with account of all rest variables of a d-dimensional theory. Qualitative dependence of quantum corrections on parameters of the classical systems is also evaluated for a much broader class of potentials u(x) = b²f(bx) + C with b and C as arbitrary real constants.

We study a nonlinear nuclear equation of state in the framework of a relativistic mean field theory. We investigate the possible thermodynamic instability in a warm and dense asymmetric nuclear medium where a phase transition from nucleonic matter to resonance dominated Δ matter can take place. Such a phase transition is characterized by both mechanical instability (fluctuations on the baryon density) and by chemical-diffusive instability (fluctuations on the isospin concentration) in asymmetric nuclear matter. Similarly to the liquid-gas phase transition, the nucleonic and the Δ-matter phase have a different isospin density in the mixed phase. In the liquid-gas phase transition, the process of producing a larger neutron excess in the gas phase is referred to as isospin fractionation. A similar effects can occur in the nucleon-Δ matter phase transition due essentially to a negative Δ-particles excess in asymmetric nuclear matter. In this context, we investigate also the effects of power law effects, due to the possible presence of nonextensive statistical mechanics effects.

We examine general statements in the Wronskian representation of Darboux transformations for plane zero-range potentials. Such expressions naturally contain scattering problem solution. We also apply Abel theorem to Wronskians for differential equations and link it to chain equations for Darboux transforms to fix conditions for further development of the underlying distribution concept. Moutard transformations give a convenient insight into the problem that allows one to formulate general assertions and give complete description of a point potential in a limit of long waves.

We represent the Benney system of dispersionless hydrodynamic equations as NLS type infinite system of equations with quantum potential. We show that negative dispersive deformation of this system is an integrable system including vector generalization of Resonant NLS and 2+1 dimensional nonlocal Resonant NLS. We obtain bilinear form and soliton solutions in these systems and find the resonant character of soliton interaction. Equivalent vector Broer-Kaup system and non-local 2+1 dimensional nonlocal Broer-Kaup equation are constructed.

A system consisting of an arbitrary number of particles of equal masses interacting via an arbitrary potential of homogeneity degree −2 and confined by an isotropic harmonic potential has the property of sustaining undamped isochronous compressional oscillations, as has been shown earlier. In this paper, we review and generalize this finding, and also the concept of thermodynamic equilibrium for such systems. It turns out that these compressional oscillations are adiabatic, and that correspondingly, the temperature varies when the size of the system does (in the specific case stated above, this dependence is one of inverse proportionality). It is also shown that some of these results extend to the quantal case.

The aim of this paper is to introduce a new category of manifolds, called Haantjes manifolds, and to show their interest by a few selected examples.

In this paper we review some results about the theory of integrable dispersionless PDEs arising as commutation condition of pairs of one-parameter families of vector fields, developed by the authors during the last years. We review, in particular, the basic formal aspects of a novel Inverse Spectral Transform including, as inverse problem, a nonlinear Riemann – Hilbert (NRH) problem, allowing one i) to solve the Cauchy problem for the target PDE; ii) to construct classes of RH spectral data for which the NRH problem is exactly solvable, corresponding to distinguished examples of exact implicit solutions of the target PDE; iii) to construct the longtime behavior of the solutions of such PDE; iv) to establish in a simple way if a localized initial datum breaks at finite time and, if so, to study analytically how the multidimensional wave breaks. We also comment on the existence of recursion operators and Backlünd – Darboux transformations for integrable dispersionless PDEs.

The nonlinear dynamics of the concentric shallow water waves is described by means of the cylindrical Korteweg-de Vries equation, often referred to as the concentric Korteweg-de Vries equation (cKdVE). By using the mapping that transforms a cKdVE into the standard one – hereafter also referred to as the planar Korteweg-de Vries equation (KdVE) – the spatiotemporal evolution of a cylindrical surface water wave, corresponding to a tilted cylindrical bright soliton, is described. The usual representation of a tilted soliton is 'non-physical'; here the cylindrical coordinate and the retarded time play the role of time-like and space-like variables, respectively. However, we show that, when we express such analytical solution of the cKdVE in the appropriate representation in terms of the two horizontal space coordinates, say X and Y, and the 'true' time, say T, this non-physical character disappears. The analysis is then carried out numerically to consider the surface water wave evolution corresponding to initially localized structures with standard boundary conditions, such as bright soliton, Gaussian and Lorentzian profiles. A comparison among those profiles is finally presented.

The Skyrme-Faddeev model admits exact analytical non localized solutions, which describe magnetic domain wall solutions when multivalued singularities appear or, differently, always regular periodic nonlinear waves, which may degenerate into linear spin waves or solitonic structures. Here both classes of solutions are derived and discussed and a general discusssion about the existence of integrable subsectors of the model is addressed.

If t_n are the heights of the Riemann zeros 1/2 + it_n, an old idea, attributed to Hilbert and Polya [6], stated that the Riemann hypothesis would be proved if the t_n could be shown to be eigenvalues of a self-adjoint operator. In 1986 Berry [1] conjectured that t_n could instead be the eigenvalues of a deterministic quantum system with a chaotic classical counterpart and in 1999 Berry and Keating [3] proposed the Hamiltonian H = xp, with x and p the position and momentum of a one-dimensional particle, respectively. This was proven not to be the correct Hamiltonian since it yields a continuum spectrum [23] and therefore a more general Hamiltonian H = w(x)(p + ℓ²_p/p) was proposed [25], [4], [24] and different expressions of the function w(x) were considered [25], [24], [16] although none of them yielding exactly t_n. We show that the quantization by means of Lie and Noether symmetries [18], [19], [20], [7] of the Lagrangian equation corresponding to the Hamiltonian H yields straightforwardly the Schrödinger equation and clearly explains why either the continuum or the discrete spectrum is obtained. Therefore we infer that suitable Lie and Noether symmetries of the classical Lagrangian corresponding to H should be searched in order to alleviate one of Berry's quantum obsessions [2].

For arbitrary hydrodynamic flow in circular annulus we introduce the two circle theorem, allowing us to construct the flow from a given one in infinite plane. Our construction is based on q-periodic analytic functions for complex potential, leading to fixed scale-invariant complex velocity, where q is determined by geometry of the region. Self-similar fractal structure of the flow with q-periodic modulation as solution of q-difference equation is studied. For one point vortex problem in circular annulus by fixing singular points we find solution in terms of q-elementary functions. Considering image points in complex plane as a phase space for qubit coherent states we construct Fibonacci and Lucas type entangled N-qubit states. Complex Fibonacci curve related to this construction shows reach set of geometric patterns.

The purpose of this research is to investigate to what extend application of novel method of complete bifurcation groups to the analysis of global dynamics of piecewise-smooth hybrid systems enables one to highlight new nonlinear effects before periodic and chaotic regimes. Results include the construction of complete one and two-parameter bifurcation diagrams, detection of various types of bifurcation groups and investigation of their interactions, localization of rare attractors, and the investigation of different principles of birth of chaotic attractors. Effectiveness of the approach is illustrated in respect to one of the most widely used switching systems-boost converter under current mode control operating in continuous current mode.

We consider the topologically nontrivial phase states and the corresponding topological defects in the SU(3) d-dimensional quantum chromodynamics (QCD). The homotopy groups for topological classes of such defects are calculated explicitly. We have shown that the three nontrivial groups are _π3SU(3) = , _π5SU(3) = , and _π6SU(3) = ₆ if 3 ≤ d ≤ 6. The latter result means that we are dealing exactly with six topologically different phase states. The topological invariants for d=3,5,6 are described in detail.

An extension of a recent method is applied in order to construct new explicit exact solutions for a system of coupled KdV-like equations.

In this paper, using a recent approach for finding conservation laws, based on Lie symmetries, we establish the conservation laws for a model admitting quasi self-adjoint equations.

We present a technique based on extended Lax Pairs to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies. As illustrative examples, we consider generalized KdV equations, and three variants of generalized MKdV equations. It is demonstrated that the technique yields Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painlevé Test, Bell Polynomials, and various similarity methods. Some solutions are also presented for the generalized KdV equation derived here by the use of the Painlevé singular manifold method. Current and future work is centered on generalizing other integrable hierarchies of NLPDEs similarly, and deriving various integrability properties such as solutions, Backlund Transformations, and hierarchies of conservation laws for these new integrable systems with variable coefficients.

The article is concerned with the study of asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value — boundary problem on a finite interval, with constant boundary conditions. Since these equations take a dissipation into account, it is naturally to presuppose that any initial profile will evolve to an invariant time-independent solution with the same boundary values. Yet the answer happens to be slightly more complex. There are three possibilities: the initial profile may regularly decay to an invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or, exotically, an asymptotic limit is a 'frozen multi-oscillation' piecewise-differentiable solution, composed of different smooth invariant solutions.

Given a n–dimensional compact Riemannian manifold (M, g) with n ≥ 5, we consider the following semi-linear elliptic equation:

where the functions a, b and h are in suitable Lebesgue spaces, 2 < q < N and λ > 0 a real parameter, f is a smooth positive function and the operator P_g is coercive. Under some additional conditions, we obtain results concering the existence of strong solutions of the above equation in H²₂ (M).

This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to become essentially a tau function of the 1D Toda hierarchy. In the same sense, the modified model turns out to be related to the Ablowitz-Ladik hierarchy. These facts are explained with the aid of a free fermion system, fermionic expressions of the partition functions, algebraic relations among fermion bilinears and vertex operators, and infinite matrix representations of those operators.

We show that the notion of generalized Lenard chains allows to formulate in a natural way the theory of multi-separable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function and by a Nijenhuis tensor defined on a symplectic manifold guarantees the separation of variables. As an application, we construct such a chain for the case I of the classical Smorodinsky–Winternitz model.

Equations of motion for the D0–brane on AdS₄ × ³ superbackground are shown to be classically integrable by extending the argument previously elaborated for the massless superparticle model.

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)×U(n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.

For kink–antikink scattering within the φ⁴ non-linear field theory in one space and one time dimension resonance type configurations emerge when the relative velocity between kink and antikink falls below a critical value. It has been conjectured that the vibrational excitation of the kink would be the source for these resonances because (simplified) collective coordinate calculations, that emphasized on this excitation, qualitatively reproduced those resonances. Surprisingly a numerical study in the ϕ⁶ field theory also exhibited such resonances even though it does not contain the vibrational excitation. To explore this contradiction we start from the working hypothesis that in either model any collective coordinate ansatz which includes a degree of freedom similar to the vibrational excitation leads to resonances, regardless of whether or not this mode emerges as a solution to the (linearized) field equations. To this end we compare numerical results in the φ⁴ and ϕ⁶ models that arise from the full set of partial differential equations to those from the ordinary differential equations for a collective coordinate ansatz. An inaccuracy in literature formulas for the collective coordinate approach in the φ⁴ model requires to revisit those calculations.

Kodama and his colleagues presented a classification theorem for exact soliton solutions of the quasi-two-dimensional Kadomtsev-Petviashvili (KP) equation. The classification theorem is related to non-negative Grassmann manifold, Gr(N, M) that is parameterized by a unique chord diagram based on the derangement of the permutation group. The cord diagram can infer the asymptotic behavior of the solution with arbitrary number of line solitons. Here we present the realization of a variety of the KP soliton formations in the laboratory environment. The experiments are performed in a water tank designed and constructed for precision experiments for long waves. The tank is equipped with a directional-wave maker, capable of generating arbitrary-shaped multi-dimensional waves. Temporal and spatial variations of water-surface profiles are captured using the Laser Induces Fluorescent method – a nonintrusive optical measurement technique with sub-millimeter precision. The experiments yield accurate anatomy of the KP soliton formations and their evolution behaviors. Physical interpretations are discussed for a variety of KP soliton formations predicted by the classification theorem.

This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [1]. We construct the generating function of integrals of motion for the quantum Gaudin model with twisted boundary conditions (the master T-operator) and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. This implies that zeros of eigenvalues of the master T-operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.