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We solved the Poisson equations, obtaining their exact solution in elementary functions for the rotation matrix of a free asymmetrical body with angular velocity vector lying on separatrices. This allows us to discuss the temporal evolution of Dzhanibekov's nut directly in the laboratory system, where it is observed. The rotation matrix depends on two parameters with clear physical interpretation as a frequency and a damping factor of the solution. Qualitative analysis of the solution shows that it properly describes a single-jump Dzhanibekov effect. • The exact solution to Poisson equations in elementary functions for the rotation matrix corresponding to separatrices is found. • The rotation matrix depends on two parameters: frequency and a damping factor of the solution. • A single-jump Dzhanibekov effect is described in the laboratory system, where it is observed. [ABSTRACT FROM AUTHOR]

Hamiltonian operators for partial differential equations are ubiquitous in mathematical models of theoretical and applied physics. In this paper the new Reduce package CDE for computations with Hamiltonian operators is presented. CDE can verify the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators, and it can compute the Lie derivative of a Hamiltonian operator with respect to a vector field. More generally, it can compute with (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of mathematical or theoretical physics. [ABSTRACT FROM AUTHOR]

We consider the first member of an extended modified Volterra lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the modified Korteweg-de Vries and nonlinear Schrödinger equations. [ABSTRACT FROM AUTHOR]

From a super extension of the Wadati, Konno and Ichikawa scheme for integrable systems and using a osp(1,2) valued connection 1-form we obtain super generalizations for the Short Pulse equation as well for the Elastic Beam equation. [ABSTRACT FROM AUTHOR]

Using a quasideterminant Darboux transformation matrix, we construct soliton solutions of nonlinear integrable discrete and semi-discrete principal chiral equations (PCEs). A Darboux transformation is defined for the matrix solutions of the discrete PCE in terms of matrix solutions to the Lax pair. The solutions are expressed in terms of quasideterminants. By taking continuum limit of one independent discrete variable, we also compute quasideterminant solutions of semi-discrete PCEs. Explicit expressions of one and two soliton solutions of the discrete PCE are obtained from a seed solution by using properties of quasideterminants. It has been shown that the soliton solutions of the discrete system reduce to those of semi-discrete and usual continuum PCEs by applying appropriate limits. [ABSTRACT FROM AUTHOR]

By means of transformations to nonlinear Klein–Gordon equations, we show that a generalized short pulse equation is integrable in two (and, most probably, only two) distinct cases of its coefficients. The first case is the original short pulse equation (SPE). The second case, which we call the single-cycle pulse equation (SCPE), is a previously overlooked scalar reduction of a known integrable system of coupled SPEs. We get the Lax pair and bi-Hamiltonian structure for the SCPE and show that the smooth envelope soliton of the SCPE can be as short as only one cycle of its carrier frequency. [ABSTRACT FROM AUTHOR]

We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems. Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously. [ABSTRACT FROM AUTHOR]

We obtain 21 two-dimensional natural Hamiltonian systems with sextic invariants, which are polynomials of the sixth order in momenta. Following to Bertrand, Darboux, and Drach these results of the standard brute force experiments can be applied to construct a new mathematical theory. • Discussion of natural Hamiltonian systems with a position-dependent effective mass having cubic invariants. • Construction of two-dimensional integrable systems with sextic invariants. • Preparation to classification of bi-Hamiltonian systems with six-order integrals of motion. [ABSTRACT FROM AUTHOR]

We implement an algorithm for the computation of Schouten bracket of weakly nonlocal Hamiltonian operators in three different computer algebra systems: Maple, Reduce and Mathematica. This class of Hamiltonian operators encompass almost all the examples coming from the theory of (1+1)-integrable evolutionary PDEs. Program Title: Jacobi (Maple), CDE module cde_weaklynl.red (Reduce, official distribution), nlPVA (Mathematica) CPC Library link to program files: https://doi.org/10.17632/synmrvr74g.1 Developer's repository link: https://gdeq.org/Weakly%5fnonlocal%5fPoisson%5fbrackets Licensing provisions: BSD 2-clause Programming language: Maple, Reduce (Rlisp), Mathematica Supplementary material: Example program files in the three languages (Maple, Reduce, Mathematica) Nature of problem: Calculating the Jacobi identity for weakly nonlocal Poisson brackets Solution method: Bringing the Jacobi identity to a canonical form Additional comments including restrictions and unusual features: Use a 2020 Maple or Mathematica version, or a 2021 Reduce snapshot [ABSTRACT FROM AUTHOR]

Abstract: We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, H, is zero we reduce this dynamical system to the motion of a particle in a one-dimensional time-independent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincaré sections of this system provide strong evidence that the motion remains bounded when and is typically quasi-periodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics. [Copyright &y& Elsevier]

Abstract: In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines. [Copyright &y& Elsevier]

Consider the Dirac operator L with the potentials u=u(x,t) and v=v(x,t) satisfying an arbitrarily taken system of two nonlinear evolution equations of the form ∂u/∂t=f, ∂v/∂t=g, the right-hand sides of which are rather smooth functions x, u, v and their derivatives with respect to x up to a certain finite order k0⩾0. It is shown that the evolution of the S-matrix of the operator L generated by the solutions of this system rapidly decreasing as x→±∞ is described by ordinary differential equations if and only if the system of equations for the potentials u and v under consideration has the Lax representation with the operator A differential with respect to x. The method established in this paper can be applied to elucidate analogous problems concerning a possible application of other operators to solve nonlinear evolution equations by the inverse scattering method. [Copyright &y& Elsevier]

• The modified Martínez Alonso–Shabat equation u y u x z + α u x u t y − (u z + α u t) u x y = 0 is studied. • Its recursion operator is presented. • An infinite commuting hierarchy of full-fledged nonlocal symmetries is found. We study the modified Martínez Alonso–Shabat equation u y u x z + α u x u t y − (u z + α u t) u x y = 0 and present its recursion operator and an infinite commuting hierarchy of full-fledged nonlocal symmetries. To date such hierarchies were found only for very few integrable systems in more than three independent variables. [ABSTRACT FROM AUTHOR]

Among the few known rigorous results for time-dependent equilibrium correlations, important for understanding transport properties, are the Mazur bound and the Suzuki equality. The Mazur inequality gives a lower bound, on the long-time average of the time-dependent auto-correlation function of observables, in terms of equilibrium correlation functions involving conserved quantities. On the other hand, Suzuki proposes an exact equality for quantum systems. In this paper, we discuss the relation between the two results and in particular, look for the analogue of the Suzuki result for classical systems. This requires us to examine as to what constitutes a complete set of conserved quantities required to saturate the Mazur bound. We present analytic arguments as well as illustrative numerical results from a number of different systems. Our examples include systems with few degrees of freedom as well as many-particle integrable models, both free and interacting. [ABSTRACT FROM AUTHOR]

• A noncommutative generalization of AB system is studied. • Zero curvature representation, conservation laws, Darboux transformation and quasideterminant solutions have been presented. • Solutions of noncommutative AB system are expressed in terms of the solutions to noncommutative sine-Gordon and noncommutative coupled dispersionless system. A noncommutative generalization of AB (nc-AB) system is studied. The generalization preserves the integrable structure of the system by exhibiting its integrability properties like zero curvature representation, conservation laws, Darboux transformation and quasideterminant solutions. Further, soliton solutions of the nc-AB system are constructed within the framework of quasideterminants. We also show that solutions of the nc-AB system can be expressed in terms of the solutions to the noncommutative sine-Gordon (nc-SG) equation and the noncommutative coupled dispersionless (nc-CD) system. The solutions are also compared with those of the usual commutative system. [ABSTRACT FROM AUTHOR]