Integrable Systems

This book gives brief but thorough enough materials on “Kinematics”, “Dynamics” and “Relativistic mechanics” of “General physics” course. It is prepared by the department of theoretical and experimental physics of Tomsk polytechnic university. This textbook intended for the students of all science disciplines of technical universities.

In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N -soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions. periodic initial data, based on both the inverse spectral theory and algebrogeometric methods, was developed by Novikov, Dubrovin, Lax, Its, Matveev et al. . For more recent reviews on the KdV equation one may refer, for instance, to Refs. .

This are some notes on mathematical physics.

Contents:
- Introduction and basics in differential geometry
-Symplectic geometry
          -Symplectic vector space
          -Symplectic mfd
          -Symplectic structure of cotangent bundle
          -Symplectomorphism and canonical transformation
          -Darboux Theorem                                                               
-Newtonian Mechanics
          -Poisson brackets
          -Physical interpretation
        -Integrability                                             
        -Co-adjoint orbits                                         
-Poisson Manifold
        -Poisson fields
        -Poisson cohomology
        -Classical R-matrix
        -Lax Representation
-Deformation quantization (with Kontsevich's thm)

We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter, thus showing that the class of integrable (3+1)-dimensional dispersionless systems with nonisospectral Lax pairs is significantly more diverse than it appeared before. The Lax pair in question is of the type recently introduced in [A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376, arXiv:1401.2122 ].

Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schro ̈dinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schro ̈dinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schro ̈dinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions.

In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N -soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions. periodic initial data, based on both the inverse spectral theory and algebrogeometric methods, was developed by Novikov, Dubrovin, Lax, Its, Matveev et al. . For more recent reviews on the KdV equation one may refer, for instance, to Refs. .

Published version is free to read at http://rdcu.be/A7w6  The search for new integrable (3+1)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated to any pair of rational functions of one variable in general position, as established below using contact Lax pairs introduced in A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376.

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.

We propose a non-perturbative solution of N = 2 supersymmetric gauge theory in five dimensions compactified on circle of a radius R. We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations. The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with 1 R playing the rôle of the speed of light. The theory with adjoint hypermultiplet exhibits some interesting finiteness properties.

We suggest a Hamiltonian formulation for the spin Ruijsenaars-Schneider system in the trigonometric case. Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars-Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly.

We consider a general multicomponent (2+1)-dimensional long-wave–short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multisoliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one-and two-soliton solutions and studied their dynamics briefly.

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of timedependent symmetries. In particular we describe how that the finite-dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time-dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painlevé equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds. 

The dispersionless KP hierarchy is considered from the point of view of the twistor formalism. A set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied. A method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed. This method is applied for determining the orbits of solutions satisfying reduction constraints of Gelfand–Dikii type under the action of additional symmetries.

We introduce a new superintegrable Kepler-Coulomb system with non-central terms in N-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give an algebraic derivation of spectrum of the superintegrable system. We show how the so(N + 1) symmetry algebra of the N-dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure
constants involving a Casimir operator of so(N − 1) Lie algebra. We construct the quadratic algebra and the Casimir operator. We show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum.

The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lamétype to obtain reductions of the vectorial Ribaucour transformations to Egorov systems. We also show that a permutability property holds for all these transformations. As an example we apply these transformations to the Cartesian background.

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined. Mathematics Subject Classification (1991): 15A66, 53A50, 53A10, 35Q55

We consider the four-dimensional integrable Martínez Alonso-Shabat equation, and present three integrable three-dimensional reductions thereof. One of these reductions, the basic Veronese web equation, provides a new example of an integrable three-dimensional PDE.

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields.

In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.

Based on the fundamental commutator representation proposed by Cao et al. we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel'fand-Dickey hierarchy.

In this paper we study the map associating to a linear differential operator with rational coefficients its monodromy data. The operator has one regular and one irregular singularity of Poincare' rank 1. We compute the Poisson structure of the corresponding Monodromy Preserving Deformation Equations on the space of the monodromy data.

Jean-Marie Souriau developed the symplectic aspects of classical mechanics and quantum mechanics. Among his most important works, let us quote the moment(um) map (geometrization of Noether's theorem), the coadjoint action of a group on its moment space, geometric quantization, the classification of homogeneous symplectic leaves, or diffeological spaces.

This thesis is concerned with the problem of constructing surfaces of constant mean curvature with irregular ends by using the class of Heun’s Differential Equations. More specifically, we are interested in obtaining immersion of punctured Riemann spheres into three dimensional Euclidean space with constant mean curvature. These immersions can be described by a Weierstrass representation in terms of holomorphic loop Lie algebra valued 1-forms. We describe how to encode each of the differential equations in Heun’s family in the Weierstrass representation. Next, we investigate monodromy problems for each of the cases in order to ensure periodicity of all the resulting immersions. This allows us to find four families of surfaces with constant mean curvature and irregular ends. These families can be described as trinoids, cylinders, perturbed Delaunay surfaces and planes. Finally, we study some symmetry properties of these groups of surfaces.

A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presented. It is used to derive solutions of the Boyer–Finley equation (dispersionless Toda equation), which are not group invariant, and the corresponding family of explicit ultra-hyperbolic selfdual vacuum spaces.

We study integrable open boundary conditions for d(2, 1; α) 2 and psu(1, 1|2) 2 spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type IIB open superstrings ending on D-branes in the AdS 3 ×S 3 ×S 3 ×S 1 and AdS 3 ×S 3 ×T 4 supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1-and D5-brane maximal giant gravitons.

We connect the theory of orthogonal Laurent polynomials on the unit circle and the theory of Toda-like integrable systems using the Gauss–Borel factorization of a Cantero–Moral–Velázquez moment matrix, that we construct in terms of a complex quasi-definite measure supported on the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials on the unit circle and the corresponding second kind functions. We obtain Jacobi operators, 5-term recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas from this point of view in a completely algebraic way. We generalize the Cantero–Moral–Velázquez sequence of Laurent monomials, recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas in this extended context. We introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. We obtain the Lax and Zakharov–Shabat equations using the classical integrability theory tools. We explicitly derive the dynamical system associated with the coefficients of the orthogonal Laurent polynomials and we compare it with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szegő polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, we obtain the representation of the orthogonal Laurent polynomials (and their second kind functions), using the formalism of Miwa shifts in terms of τ
-functions and the subsequent bilinear equations.

In this paper, surface-activation-based nanobonding technology and its applications are described. This bonding technology allows for the integration of electronic, photonic, fluidic and mechanical components into small form-factor systems for emerging sensing and imaging applications in health and environmental sciences. Here, we describe four different nanobonding techniques that have been used for the integration of various substrates—silicon, gallium arsenide, glass, and gold. We use these substrates to create electronic (silicon), photonic (silicon and gallium arsenide), microelectromechanical (glass and silicon), and fluidic (silicon and glass) components for biosensing and bioimaging systems being developed. Our nanobonding technologies provide void-free, strong, and nanometer scale bonding at room temperature or at low temperatures (<200 °C), and do not require chemicals, adhesives, or high external pressure. The interfaces of the nanobonded materials in ultra-high vacuum and in air correspond to covalent bonds, and hydrogen or hydroxyl bonds, respectively.

Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schrödinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schrödinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schrödinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions. †

We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with m≥2 vertices. Their global Poisson structure is characterised by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case m=1 corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars-Schneider system. This generalises to new spin cases recent works on classical integrable systems in the Ruijsenaars-Schneider family.

We study the motion of the Kovalevskaya top about a fixed point, construct invariant sets (in particular, images of Liouville tori and three-dimensional isoenergetic surfaces) in the movable space of angular velocities, and classifies all possible types of those sets. A family of curves on Poisson sphere illustrates the Liouville foliation for Kovalevskaya top.

We introduce a new family of N-dimensional quantum superintegrable model consisting of double singular oscillators of type (n, N−n). The special cases (2,2) and (4,4) were previously identified as the duals of 3- and 5-dimensional deformed Kepler-Coulomb systems with u(1) and su(2) monopoles respectively.  The models are multiseparable and their wave functions are obtained in (n, N−n) double hyperspherical coordinates.  We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra Q(3) involving three generators, so(n), so(N−n)  (i.e. Q(3)⊕so(n)⊕so(N−n) ). The structure constants of the quadratic algebra themselves involve the Casimir operators of the two Lie algebras so(n) and so(N−n). Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the  superintegrable model.

Multiple orthogonality is considered in the realm of a Gauss–Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss–Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel–Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss–Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov–Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the τ-function representation of the multiple orthogonality is given.

The Grassmannian formalism of KP hierarchies is used to study geometric nets of orthogonal type and their subclass of Egorov nets. Efficient dressing methods for Cauchy propagators are provided which lead to wide families of explicit nets. Frobenius manifolds and solutions to the Witten-Dijkgraff-Verlinde-Verlinde associativity equations are also constructed.

We demonstrate separability of the Dirac equation in weakly charged rotating black hole spacetimes in all dimensions. The electromagnetic field of the black hole is described by a test field approximation, with vector potential proportional to the primary Killing vector field. It is shown that the demonstrated separability can be intrinsically characterized by the existence of a complete set of mutually commuting first order symmetry operators generated from the principal Killing-Yano tensor. The presented results generalize the results on integrability of charged particle motion and separability of charged scalar field studied in .

Two simple ways to identify and explain fake Lax pairs are provided. The two methods are complementary, one involves finding a gauge transformation which can be used to remove the associated nonlinear system's dependent variable(s) from a fake Lax pair. The second method shows that excess degrees of freedom exist in fake Lax pairs. We provide several examples to illustrate both tests. The tests proposed here can be applied to all types of Lax pairs, including scalar or matrix linear problems associated with ordinary or partial difference and/or differential systems.

The main goal of this thesis is to provide a systematic study of several integrable systems defined on complex Poisson manifolds associated to extended cyclic quivers. These spaces are particular examples of multiplicative quiver varieties of Crawley-Boevey and Shaw, for which Van den Bergh observed that they can be equipped with a Poisson bracket obtained by quasi-Hamiltonian reduction. In his approach, Van den Bergh introduced the notion of double brackets to translate the geometric quasi-Hamiltonian structure associated to these varieties directly at the level of the path algebra of the quivers. We pursue this line of thought and examine these double brackets in order to find families of algebraic elements on the path algebra of extended cyclic quivers that give rise to families of Poisson commuting functions on the corresponding multiplicative quiver varieties. This provides a way to obtain candidates for Liouville integrability, and this can be adapted to the case of degenerate integrability. For specific dimensions of these spaces, we can compute the number of functionally independent elements in each family, and conclude that we can form integrable systems. They can be written in terms of local coordinates, and be related to the trigonometric spin Ruijsenaars-Schneider system or generalisations of the latter system. As part of our construction, we also prove that their flows can be obtained by the projection method from explicit integrations performed before the quasi-Hamiltonian reduction. Another application of this work consists in describing the Poisson structure in terms of local coordinates. In particular, this allows us to prove a conjecture of Arutyunov and Frolov regarding the form of the Poisson bracket for the trigonometric spin Ruijsenaars-Schneider system.

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schroedinger operator, deep connections with special functions and with QES systems. Here we announce a complete classification of nondegenerate (i.e., 4-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in 10 variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly 10 nondegenerate potentials.

In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not.

As soon as 1993, A. Fukjiwara and Y. Nakamura have developed close links between Information Geometry and integrable system by studying dynamical systems on statistical models and completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions, Recently, Jean-Pierre Françoise has rediscovered this model. In parallel, in the framework of integrable dynamical systems and Moser isospectral deformation method, originated in the work of P. Lax, A.M. Perelomov has applied models for space of Positive Definite Hermitian matrices of order 2, considering this space as an homogeneous space. We conclude with links of Lax pairs with Souriau equation to compute coefficients of characteristic polynomial of a matrix. Main objective of this paper is to compose a synthesis of these researches on integrability and Lax pairs and their links with Information Geometry, to stimulate new developments at the interface of these disciplines.

The propagator for the 2D heat equation in an arbitrary linear space is shown to give solutions of the two-component Kadomtsev - Petviashvilii (KP) equations, also called Davey - Stewartson system. This propagator is subject to the Klein - Gordon equation and its right-derivatives are required to be of rank one, that imply that it can be expressed in terms of solutions of the Dirac equation. Large families of solutions of the two-component Kadomtsev - Petviashvilii equations are constructed in terms of solutions of the heat and Dirac equations. Particular attention is paid to the real reductions of the Davey - Stewartson type, recovering in this way the line solitons and the multidromion solutions. Moreover, new solutions to the Davey - Stewartson I are presented as massive deformations of the dromion.

We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the momenta to polynomials in the momenta. We study specializations of these actions which do preserve polynomials and also the structure of the symmetry algebras in both the classical and quantum cases. We give several examples of non-constant curvature 3rd and 4th order superintegrable systems in 2 space dimensions obtained via CCM, with some details on the structure of the symmetry algebras preserved by the transform action.

An alternative expression for the Christoffel–Darboux formula for multiple orthogonal polynomials of mixed type is derived from the LU factorization of the moment matrix of a given measure and two sets of weights. We use the action of the generalized Jacobi matrix J, also responsible for the recurrence relations, on the linear forms and their duals to obtain the result.

Inspired by the results of Jonas, Einsenhart, Demoulin, and Bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions—of pseudo-orthogonal, pseudo-symmetric, and pseudo-Egorov types—dressing transformations of the N-component KP hierarchy (described within the Grassmannian) are used to generate quadrilateral lattices and its corresponding reductions. As a byproduct we get the corresponding discrete dressing transformations; in particular, we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice.

We construct Darboux transformations for the super-symmetric KP hierarchies of Manin–Radul and Jacobian types. We also consider the binary Darboux transformation for the hierarchies. The iterations of both type of Darboux transformations are briefly discussed.