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We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity s = -\frac 12 as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin–Ono equation (BO) and exploiting the complete integrability of BO, we establish a global-in-time a priori bound on the H^s-norm of a solution to ILW for - \frac 12 < s < 0. (ii) By making use of explicit solutions, we prove that ILW is ill-posed in H^s for s < - \frac 12. Our results apply to both the real line case and the periodic case. [ABSTRACT FROM AUTHOR]

We consider linear autonomous systems of second-order differential equations that do not contain first derivatives of independent variables. Such systems are often encountered in classical mechanics. Of particular interest are cases where external forces are not potential. An important special case is given by the equations of nonholonomic mechanics linearized in the vicinity of equilibria of the second kind. We show that linear systems of this type can always be represented as Lagrange and Hamilton equations, and these equations are completely integrable: they admit complete sets of independent involutive integrals that are quadratic or linear in velocity. The linear integrals are Noetherian: they appear due to nontrivial symmetry groups. [ABSTRACT FROM AUTHOR]

A thermodynamically unstable spin glass growth model described by means of the parametrically-dependent Kardar–Parisi–Zhang equation is analyzed within the symplectic geometry-based gradient–holonomic and optimal control motivated algorithms. The finitely-parametric functional extensions of the model are studied, and the existence of conservation laws and the related Hamiltonian structure is stated. A relationship of the Kardar–Parisi–Zhang equation to a so called dark type class of integrable dynamical systems, on functional manifolds with hidden symmetries, is stated. [ABSTRACT FROM AUTHOR]

Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities. [ABSTRACT FROM AUTHOR]

In this paper we will solve an open problem raised by Manásevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial p -Laplacian. This is an Eulerâ€"Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent p other than 2, the vectorial p -Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to two-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of K n (K = C or R) . We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp. a smooth) transformation if the initial diffeomorphisms are analytic (resp. smooth). [ABSTRACT FROM AUTHOR]

We consider the nonlinear ion acoustic wave induced by orbiting charged space debris objects in the plasma environment generated at the Low Earth Orbital (LEO) region. The generated nonlinear ion acoustic wave is shown to be governed by a forced Korteweg-de Vries (fKdV) equation with the forcing function dependent on the charged space debris function. For a specific relationship between the forcing debris function and the nonlinear ion acoustic wave, the forced KdV equation turns out to be a completely integrable system; where the debris function obeys a definite nonholonomic constraint. A special exact accelerated soliton solution (velocity of the soliton changes over time whereas its amplitude remains constant) has been derived for the ion acoustic wave for the first time. On the other hand, the amplitude of the solitonic debris function varies with time, and its shape changes during the propagation. Approximate ion acoustic solitary wave solutions with time varying amplitudes and velocities have been derived for different types of weakly localized charged debris functions. Possible applications of the obtained results in space plasma physics are stated along with future directions for research. [ABSTRACT FROM AUTHOR]

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$ -dimensional torus that is without conjugate points is $C^{0}$ -integrable, that is   $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs. [ABSTRACT FROM AUTHOR]

The approach based on commutator identities for elements of associative algebras was previously effectively used to investigate -dimensional integrable systems. We develop this approach to investigate integrable hierarchies and their relations. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new function space M H θ , p whose norm is given by the ℓ p -sum of modulated H θ -norms of a given function. In particular, when θ < − 1 2 , we show that the space M H θ , p agrees with the modulation space M 2 , p (R) on the real line and the Fourier-Lebesgue space F L p (T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on R is globally well-posed in M 2 , p (R) for any p < ∞ , while the renormalized cubic NLS on T is globally well-posed in F L p (T) for any p < ∞. In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation. [ABSTRACT FROM AUTHOR]

It is proved that the Schrödinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Integrals of motion are presented. A similar statement is proved for classical dynamical systems in terms of Koopman's approach to dynamical systems. Examples of explicit reduction of quantum and classical dynamics to the family of harmonic oscillators by using direct methods of scattering theory and wave operators are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to go further into the study of the quadratic Szegő equation, which is the following Hamiltonian PDE: ..., where Π is the Szegő projector onto nonnegative modes, and J = J(u) is the complex number given by J = ∫T|u|²u. We exhibit an infinite set of new conservation laws {ℓk} which are in involution. These laws give us a better understanding of the "turbulent" behavior of certain rational solutions of the equation: we show that if the orbit of a rational solution is unbounded in some Hs, s > 1/2, then one of the ℓk's must be zero. As a consequence, we characterize growing solutions which can be written as the sum of two solitons. [ABSTRACT FROM AUTHOR]

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson–Lie algebras: A proof of the Mischenko–Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87–109.)

We consider the fifth-order modified Korteweg–de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in H s (𝕋) , s > 2 , via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math.173(2) (2008) 265–304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation. [ABSTRACT FROM AUTHOR]

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds (rcs) over Sasaki–Einstein space T1,1. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki–Einstein space T1,1 and its Calabi–Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution, the geodesic motion on the rc remains completely integrable. Instead, in the case of the deformation of the conifold, the complete integrability is lost. [ABSTRACT FROM AUTHOR]

The cosmological dynamics of a quintessence model based on real gas with general equation of state is presented within the framework of a three-dimensional dynamical system describing the time evolution of the number density, the Hubble parameter and the temperature. Two global first integrals are found and examples for gas with virial expansion and van der Waals gas are presented. The van der Waals system is completely integrable. In addition to the unbounded trajectories, stemming from the presence of the conserved quantities, stable periodic solutions (closed orbits) also exist under certain conditions and these represent models of a cyclic Universe. The cyclic solutions exhibit regions characterized by inflation and deflation, while the open trajectories are characterized by inflation in a 'fly-by' near an unstable critical point. [ABSTRACT FROM AUTHOR]

The study of higher-dimensional black holes is a subject which has recently attracted vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional Kerr-NUT-(A)dS black hole spacetimes. We start with discussion of the Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a 'seed object' which generates all these symmetries. It determines the form of the geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations. [ABSTRACT FROM AUTHOR]

We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and E6 by Goodman and Wallach at the beginning of the 1980s. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types. [ABSTRACT FROM AUTHOR]

The paper concerns a simple model of bicycle kinematics: a bicycle is represented by an oriented segment of constant length in n -dimensional space that can move in such a way that the velocity of its rear end is aligned with the segment (the rear wheel is fixed on the bicycle frame). Starting with a closed trajectory of the rear end, one obtains the two respective trajectories of the front end, corresponding to the opposite directions of motion. These two curves are said to be in the bicycle correspondence. Conjecturally, this correspondence is completely integrable; we present a number of results substantiating this conjecture. In dimension three, the bicycle correspondence is the Bäcklund transformation for the filament equation; we discuss bi-Hamiltonian features of the bicycle correspondence and its integrals. [ABSTRACT FROM AUTHOR]

The three-dimensional champagne bottle system contains no mondromy, despite being entirely composed of invariant two-dimensional champagne bottle systems, each of which posesses nontrivial monodromy. We explain where the monodromy went in the three-dimensional system, or perhaps, where it did come from in the two-dimensional system, by regarding the threedimensional system not as completely integrable, but as superintegrable (or non-commutatively integrable), and explaining the role of the singularities of its isotropic-coisotropic pair of foliations. [ABSTRACT FROM AUTHOR]

We consider differential systems obtained by coupling two Euler-Poinsot systems. The motivation to consider such systems can be traced back to the Riemann ellipsoid problem. We provide new cases for which these systems are completely integrable. We also prove that these systems either are completely integrable or have at most four functionally independent analytic first integrals. [ABSTRACT FROM AUTHOR]

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Stróżyna-Żołladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and sl2(C)-representations. [ABSTRACT FROM AUTHOR]

Consider a complex Hamiltonian system and an integral curve. In this paper, we give an effective and efficient procedure to put the variational equation of any order along the integral curve in reduced form provided that the previous one is in reduced form with an abelian Lie algebra. Thus, we obtain an effective way to check the Morales–Ramis–Simó criterion for testing meromorphic Liouville integrability of Hamiltonian systems. [ABSTRACT FROM AUTHOR]

Commonused Poincaré's method to study of qualitative behavior of polynomial systems at infinity is revisited. An effect called "loss of projectivity" in dynamical systems of even degree is accented. It is given an interpretation of Lefschetz's equation in terms of Pfaff forms and a modified Lefschetz's equation is suggested which allows to get projective extension for dynamical systems of both odd and even degrees. The constructions imply that every polynomial vector field in Rd can be extended onto projective space PRd (d = 2, 3, 4, ...) saving polynomiality. [ABSTRACT FROM AUTHOR]

A ( 1 + 1 ) -dimensional nonlinear evolution equation with Lax integrability is investigated in this paper with the aid of differential forms and exterior differentials. Equivalent definition of the Lax integrability is given. Relation between the Lax integrability and complete integrability is discussed. Geometric interpretation of the Lax equation is presented. Gauge transformation and Darboux transformation are restated in terms of the differential forms and exterior differentials. If λ I − S is a Darboux transformation between two Lax equations, the system that the matrix S should satisfy is given, where λ is the spectral parameter and I is the identity matrix. The completely integrable condition of that system is obtained according to the Frobenius theorem, and it is shown that S = H Λ H − 1 satisfies that completely integrable condition with H and Λ as two matrices. [ABSTRACT FROM AUTHOR]

In this work we study complete integrability and time-reversibility of three dimensional polynomial systems with a singularity at the origin at which the linear part has a zero eigenvalue and a pair of eigenvalues that are the negatives of one another. An approach to finding systems within a given parametric family admitting two independent analytic local first integrals in a neighborhood of the origin is proposed and applied to a family of cubic systems. [ABSTRACT FROM AUTHOR]

We construct explicitly the constants of motion for geodesics in the five-dimensional Sasaki-Einstein spaces . To carry out this task, we use the knowledge of the complete set of Killing vectors and Killing-Yano tensors on these spaces. In spite of the fact that we generate a multitude of constants of motion, only five of them are functionally independent implying the complete integrability of geodesic flow on spaces. In the particular case of the homogeneous Sasaki-Einstein manifold the integrals of motion have simpler forms and the relations between them are described in detail. [ABSTRACT FROM AUTHOR]

The author returns to constructing the nonlinear mathematical model of a plane-parallel impact of a medium on a rigid body with allowance for the dependence of the arm of force on the reduced angular velocity of the body (a kind of Strouhal number). In this case, the arm of the force is also a function of the angle of attack. As was shown by the processing of the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of a plane model of the interaction of a rigid body with a medium revealed new cases of the complete integrability in elementary functions, which made it possible to find qualitative analogies between the motion of free bodies in a resisting medium and vibrations of partly fixed bodies in a flow of an oncoming medium. Phase patterns obtained in the analysis of the nonlinear model of the impact of the medium are compared with real Karman's vortex streets. [ABSTRACT FROM AUTHOR]

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $$T^*\mathbb {T}^n$$ of the $$n$$ -dimensional torus that have no conjugate points are $$C^0$$ integrable, i.e. $$T^*\mathbb {T}^n$$ is $$C^0$$ foliated by a family $$\mathcal {F}$$ of invariant $$C^0$$ Lagrangian graphs. Assuming that the Hamiltonian is $$C^\infty $$ , we prove that there exists a $$G_\delta $$ subset $$\mathcal {G}$$ of $$\mathcal {F}$$ such that the dynamics restricted to every element of $$\mathcal {G}$$ is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $$C^0$$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish. [ABSTRACT FROM AUTHOR]