Nonlinear Dynamics on the Plane and Integrable Hierarchies of Infinitesimal Deformations

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous     -equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources), is described by Hamilton–Jacobi-type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical     -dressing method.     

Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian

We consider cohomological and Poisson structures associated with the special tautological subbundles $TB_{W_{1,2, \ldots n} }$ for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of $TB_{W_{1,2, \ldots n} }$ are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev-Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in $TB_{W_{1,2, \ldots n} }$ can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in $TB_{W_{\hat S} }$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures.     

Reductions of the Dispersionless KP Hierarchy

We present a method for constructing the S-function based on a system of first-order differential equations and use it to analyze reductions of dispersionless integrable hierarchies.     

Bifurcation of a Whitney-Smooth Family of Coisotropic Invariant Tori of a Hamiltonian System under Small Deformations of a Symplectic Structure

We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space.     

Elliptic parameterization of Pfaff integrable hierarchies in the zero-dispersion limit

We show that the dispersionless limits of the Pfaff–KP (also known as the DKP or Pfaff lattice) and the Pfaff–Toda hierarchies admit a reformulation in terms of elliptic functions. In the elliptic form, they look like natural elliptic deformations of the respective dispersionless KP and two-dimensional Toda hierarchies.     

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.     

Integrable Chains and Hierarchies of Differential Evolution Equations

A class of integrable differential–difference systems is constructed based on auxiliary linear equations defined on sequences of Zakharov–Shabat formal dressing operators. We show that the auxiliary equations are compatible with the evolution equations for the Kadomtsev–Petviashvili (KP) hierarchy. The investigation results are used to elaborate a modified version of Krichever rational reductions for KP hierarchies.     

Mechanical systems related to the Schrödinger spectral problem

We present a variety of integrable mechanical systems which are embedded into the KdV and the Harry Dym hierarchies of soliton equations as their restricted flows. Integrable structures of these systems are systematically derived from the properties of the underlying hierarchies of equations. It is illustrated by the key example of the Garnier system which describes the motion of a particle in a quartic ‘wine bottle’ potential.     

Symmetry constraints for real dispersionless Veselov-Novikov equation

Symmetry constraints for dispersionless integrable equations are discussed. It is shown that under symmetry constraints, the dispersionless Veselov-Novikov equation is reduced to the (1+1)-dimensional hydrodynamic-type systems. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.     

On a Lagrangian Reduction and a Deformation of Completely Integrable Systems

We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm \(H^1\) in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered, and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them, we found two important equations, the Camassa–Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation.     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

Quasiconformal Mappings and Solutions of the Dispersionless KP Hierarchy

A formalism for studying dispersionless integrable hierarchies is applied to the dispersionless KP (dKP) hierarchy. We relate this formalism to the theory of quasiconformal mappings on the plane and present some classes of explicit solutions of the dKP hierarchy.     

A generalization of the coupled integrable dispersionless equations

The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current‐fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine‐Gordon equation and the two‐dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. N‐soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one‐soliton and two‐soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.     

Classifying Integrable Egoroff Hydrodynamic Chains

We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case. We find (2+1)-dimensional equations that are not just generalizations of the already known Khokhlov–Zabolotskaya and Boyer–Finley equations but are much more involved. These equations are parameterized by theta functions and by solutions of the Chazy equations. We obtain analogues of the dispersionless Hirota equations.     

Semiclassical $$\bar \partial $$ -Method: Generating Equations for Dispersionless Integrable Hierarchies

We use the semiclassical -dressing method to derive compact generating equations for dispersionless hierarchies. The considered illustrative examples are the dispersionless Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies.     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.     

A note on the generalized symmetries of lattice hierarchies

We show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential-delay equations. All members of these hierarchies of lattice and differential-delay equations commute. It is thus seen that differential-delay hierarchies provide a broader context within which to place lattice hierarchies.     

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

We construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.     

On the nesting of Painlevé hierarchies: A Hamiltonian approach

We consider the phenomenon whereby two different Painlevé hierarchies, related to the same hierarchy of completely integrable equations, are such that solutions of one member of one of the Painlevé hierarchies are also solutions of a higher-order member of the other Painlevé hierarchy. An explanation is given in terms of the Hamiltonian structures of the related underlying completely integrable hierarchies, and is sufficiently generally formulated so as to be applicable equally to both continuous and discrete Painlevé hierarchies. Special integrals of a further Painlevé hierarchy related by Bäcklund transformation to the other Painlevé hierarchy mentioned above can also be constructed. Examples of the application of this approach to Painlevé hierarchies related to the Korteweg–de Vries, dispersive water wave, Toda and Volterra integrable hierarchies are considered. Our results provide further evidence of the importance of the underlying structures of related completely integrable hierarchies in understanding the properties of Painlevé hierarchies.