An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Finite-type integrable geometric structures

In this paper, we consider finite-type geometric structures of arbitrary order and solve the integrability problem for these structures. This problem is equivalent to the integrability problem for the corresponding G-structures. The latter problem is solved by constructing the structure functions for G-structures of order ≥1. These functions coincide with the well-known ones for the first-order G-structures, although their constructions are different. We prove that a finite-type G-structure is integrable if and only if the structure functions of the corresponding number of its first prolongations are equal to zero. Applications of this result to second-and third-order ordinary differential equations are noted. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

A characterization of strongly measurable Kurzweil–Henstock integrable functions and weakly continuous operators

We give necessary and sufficient conditions for the Kurzweil–Henstock integrability of functions given by , where x_n belong to a Banach space and the sets (E_n)_n are measurable and pairwise disjoint. Also weakly completely continuous operators between Banach spaces are characterized by means of scalarly Kurzweil–Henstock integrable functions.     

A new class of integrable systems

This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=¹/₂(x₁²+x₂²) +V(y₁,y₂) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y₁,y₂)=¹/₂(a₁y₁²+a₂y₂²) + ¹/₄b₁y₁⁴ + ¹/₄b₂y₂⁴ + ¹/₂b₃y₁²y₂² + ?_k=1³g_k(y₁²+y₂²) ^k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a₁,a₂,b₁,b₂,b₃,g₁,g₂,g₃) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .     

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

In this paper the problem of classification of integrable natural Hamiltonian systems with n degrees of freedom given by a Hamilton function, which is the sum of the standard kinetic energy and a homogeneous polynomial potential V of degree k > 2, is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential V is not generic if it admits a nonzero solution of equation V′(d) = 0. The existence of such a solution gives very strong integrability obstructions obtained in the frame of the Morales-Ramis theory. This theory also gives additional integrability obstructions which have the form of restrictions imposed on the eigenvalues (λ ₁, …, λ _n ) of the Hessian matrix V″(d) calculated at a nonzero d ∈ ℂ ⁿ satisfying V′(d) = d. In our previous work we showed that for generic potentials some universal relations between (λ ₁, …, λ _n ) calculated at various solutions of V′ (d) = d exist. These relations allow one to prove that the number of potentials satisfying the necessary conditions for the integrability is finite. The main aim of this paper was to show that relations of such forms also exist for nongeneric potentials. We show their existence and derive them for the case n = k = 3 applying the multivariable residue calculus. We demonstrate the strength of the results analyzing in details the nongeneric cases for n = k = 3. Our analysis covers all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for n = k = 3, thanks to this analysis, a three-parameter family of potentials integrable or superintegrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.      

A note on degenerate corank-one singularities of integrable Hamiltonian systems

We prove that, in a neighborhood of a corank-1 singularity of an analytic integrable Hamiltonian system with n degrees of freedom, there is a locally-free analytic symplectic \Bbb T^n-1 {\Bbb T}^{n-1} -action which preserves the moment map, under some mild conditions. This result allows one to classify generic degenerate corank-one singularities of integrable Hamiltonian systems. It can also be applied to the study of (non)integrability of perturbations of integrable systems.     

Five constructions of integrable dynamical systems connected with the Korteweg-de Vries equation

Two countable sets of integrable dynamical systems which turn into the Korteweg-de Vries equation in a continous limit are constructed. The integrability of the dynamics of the scattering matrix entries for these systems is proved and an integrable reduction in the finitedimensional case is pointed out. A construction of the integrable dynamical systems connected with the simple Lie algebras and generalizing the discrete kdV equation is presented. Two general constructions of differential and integro-differential equations (with respect to time t) possessing a countable set of first integrals are found. These equations admit the Lax representation in some infinite-dimensional subalgebras of the Lie algebra of integral operators on an arbitrary manifold M ⁿ with measure . A construction of matrix equations having a set of attractors in the space of all matrix entries is given.     

Discrete nonlinear hyperbolic equations. Classification of integrable cases

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ?². The fields are associated with the vertices and an equation of the form Q(x ₁, x ₂, x ₃, x ₄) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ? ^N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.     

High‐contrast homogenization of linear systems of partial differential equations

We give some integrability conditions for the coefficients of a sequence of elliptic systems with varying coefficients in order to obtain the stability for homogenization. In the case of equations, it is well known that equi‐integrability and bound in L¹ are enough for this purpose; however, this is based on the maximum principle, and then, it does not work for systems. Here, we use an extension of the Murat–Tartar div‐curl lemma because of M. Briane, J. Casado‐Díaz, and F. Murat in order to obtain the stability by homogenization for systems uniformly elliptic, with bounded coefficients in , with N the dimension of the space. We also show that a weaker ellipticity condition can be assumed, but then, we need a stronger integrability for the coefficients. Copyright © 2016 John Wiley & Sons, Ltd.     

Modulated waves in a semiclassical continuum limit of an integrable NLS chain

A one‐dimensional integrable lattice system of ODEs for complex functions Q_n(τ) that exhibits dispersive phenomena in the phase is studied. We consider wave solutions of the local form Q_n(τ) ∼ qexp(i(kn + ωτ + c)), in which q, k, and ω modulate on long time and long space scales t = ετ and x = εn. Such solutions arise from initial data of the form Q_n(0) = q(nε) exp(iϕ(nε)/ε), the phase derivative ϕ′ 0 giving the local value of the phase difference k. Formal asymptotic analysis as ε → 0 yields a first‐order system of PDEs for q and ϕ′ as functions of x and t. A certain finite subchain of the discrete system is solvable by an inverse spectral transform. We propose formulae for the asymptotic spectral data and use them to study the limiting behavior of the solution in the case of initial data |Q_n| < 1, which yield hyperbolic PDEs in the formal limit. We show that the hyperbolic case is amenable to Lax‐Levermore theory. The associated maximization problem in the spectral domain is solved by means of a scalar Riemann‐Hilbert problem for a special class of data for all times before breaking of the formal PDEs. Under certain assumptions on asymptotic behaviors, the phase and amplitude modulation of the discrete systems is shown to be governed by the formal PDEs. Modulation equations after breaking time are not studied. Full details of the WKB theory and numerical results are left to a future exposition. © 2000 John Wiley & Sons, Inc.     

The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes

For a wide class of local martingales (M _t ) there is a default function, which is not identically zero only when (M _t ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup _s≤t M _s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times. Received: 6 August 1996 / Revised version: 27 July 1998     

Method for Solving the Multidimensional n‐Wave Resonant Equations and Geometry of Generalized Darboux‐Manakov‐Zakharov Systems

The intrinsic geometric properties of generalized Darboux‐Manakov‐Zakharov systems of semilinear partial differential equations (1) for a real‐valued function u(x₁, …, x_n) are studied with particular reference to the linear systems in this equation class. System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive tools for explicitly constructing involutive systems of the form (1) , essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multidimensional n‐wave resonant interaction system and its modified version. For each integer n≥ 3 and nonnegative integer k, our procedure constructs solutions of the n‐wave resonant interaction system depending on at least k arbitrary functions each of one variable. The construction of these solutions relies only on differentiation, linear algebra, and the solution of ordinary differential equations.     

Almost integrable evolution equations

We present a 2-component equation with exactly two nontrivial generalized symmetries, a counterexample to Fokas' conjecture that equations with as many symmetries as components are integrable. Furthermore we prove the existence of infinitely many evolution equations with finitely many symmetries. We introduce the concept of almost integrability to describe the situation where one has a finite number of symmetries. The symbolic calculus of Gel'fand-Dikiî andp-adic analysis are used to prove our results.     

A note on the behavior of integrable functions at infinity

We discuss a scale of necessary conditions for the integrability of a function f:[a,∞)→R, based on the concept of limit in density.     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90)     

Singular Manakov flows and geodesic flows on homogeneous spaces of SO(N)

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces SO(n)/SO(k₁) ×⋯× SO(k _r ), for any choice of k ₁,…,k _r , k ₁ + ⋯ + k _r ⩽ n. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented. Also, the proof of integrability of the SO(n)-invariant Einstein metrics on SO(k ₁ + k ₂ + k ₃)/SO(k ₁) × SO(k ₂) × SO(k ₃) and on the Stiefel manifolds V (n, k) = SO(n)/SO(k) is given.     

Strongly nonlinear modal equations for nearly integrable PDEs

Summary The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations ofN-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to anyN-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation orderN. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.