Hierarchy Of Ordinary Differential Equations Research Articles

Multicomponent Painlevé ODEs and related nonautonomous KdV stationary hierarchies

AbstractFirst, starting from two hierarchies of autonomous Stäckel ordinary differential equations (ODEs), we reconstruct the hierarchy of Korteweg de Vries (KdV) stationary systems. Next, we deform considered autonomous Stäckel systems to nonautonomous Painlevé hierarchies of ODEs. Finally, we reconstruct the related nonautonomous KdV stationary hierarchies from respective Painlevé systems.

Data-driven optimal closures for mean-cluster models: Beyond the classical pair approximation.

This study concerns the mean-clustering approach to modeling the evolution of lattice dynamics. Instead of tracking the state of individual lattice sites, this approach describes the time evolution of the concentrations of different cluster types. It leads to an infinite hierarchy of ordinary differential equationswhich must be closed by truncation using a so-called closure condition. This condition approximates the concentrations of higher-order clusters in terms of the concentrations of lower-order ones. The pair approximation is the most common form of closure. Here, we consider its generalization, termed the "optimal approximation," which we calibrate using a robust data-driven strategy. To fix attention, we focus on a recently proposed structured lattice model for a nickel-based oxide, similar to that used as cathode material in modern commercial Li-ion batteries. The form of the obtained optimal approximation allows us to deduce a simple sparse closure model. In addition to being more accurate than the classical pair approximation, this "sparse approximation" is also physically interpretable which allows us to a posteriori refine the hypotheses underlying construction of this class of closure models. Moreover, the mean-cluster model closed with this sparse approximation is linear and hence analytically solvable such that its parametrization is straightforward, although it offers a good approximation of the actual time evolution of the cluster concentrations on short timescales only. On the other hand, parametrization of the mean-cluster model closed with the pair approximation is shown to lead to an ill-posed inverse problem.

Evolution of interface singularities in shallow water equations with variable bottom topography

AbstractWave front propagation with nontrivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of nonsmooth initial data is examined, and, in particular, the splitting of singular points and their short time behavior is described. In the opposite limit of longer times, the local analysis of wave fronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so‐called “physical” and “nonphysical” vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time‐dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for nonphysical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, whereas for physical vacuums, the hierarchy is nonrecursive, fully coupled, and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, whereas the latter is shown to develop nonstandard velocity shocks instantaneously. Polynomial bottom topographies simplify the hierarchy, as they contribute only a finite number of inhomogeneous forcing terms to the equations in the recursion relations. However, we show that truncation to finite‐dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case, the system's evolution can reduce to, and is completely described by, a low‐dimensional dynamical system for the time‐dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and, in particular, governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self‐similar solution class is introduced and studied to illustrate via closed‐form expressions the general results.

Bäcklund transformations for a new extended Painlevé hierarchy

In a recent paper we introduced an extended second Painlevé hierarchy and studied its properties. The approach developed in order to derive these results is widely applicable. Here we use it to obtain a second example of an extended Painlevé hierarchy. We also give results on Bäcklund transformations, auto-Bäcklund transformations and other properties of this and related hierarchies of ordinary differential equations, as well as on the nesting of equations whereby we obtain relations between systems of different orders but of the same form.

A special solution technique: Further extensions

In a recent paper we gave a technique for identifying special solutions of hierarchies of ordinary differential equations (ODEs) and difference equations. In particular, we considered the phenomenon whereby members of one Painlevé hierarchy define solutions of members of a different Painlevé hierarchy, where both are related to the same (at least) bi-Hamiltonian completely integrable partial differential equation (PDE) or lattice hierarchy; we also considered the question of special integrals of further related Painlevé hierarchies. Here we consider three extensions of this approach. First of all, we consider the case of ODE hierarchies where the corresponding PDE hierarchy is only mono-Hamiltonian. Second, we show that our approach can also be extended in order to provide special solutions of PDEs. Our third extension is to the non-Hamiltonian case, where we consider hierarchies of ODEs related to the Burgers hierarchy. We also make the observation that our results can be phrased quite generally, and the systems of equations dealt with do not in fact need to be integrable.

Complex manifolds for the Euler equations: a hierarchy of ODEs and the case of vanishing angle in two dimensions

This paper considers the two-dimensional Euler equation for complex spatial variables and two complex modes in the initial condition. A hierarchy of third-order ordinary differential equations (ODEs) is used to study the location and structure of the complex singular manifold for short times. The system has two key parameters, the ratio η of the wave numbers of the two modes, and the angle ϕ between the two wave vectors. Using this hierarchy for the case ϕ=π/2 the results of earlier authors (Pauls et al 2006 Physica D 219 40–59) are reproduced numerically. To make analytical progress, the paper considers the limit ϕ→ 0 in which the wave vectors become parallel, rescaling time also. By considering the limiting behaviour of the ODE hierarchy, an asymptotic framework is set up that describes the geometry of the singular manifold and local behaviour of vorticity in this limiting case ϕ=0 of parallel modes. In addition, the hierarchy of ODEs can be solved analytically, order by order, in the parallel case using computer algebra. This is used to confirm the asymptotic theory and to give evidence for a scaling exponent β=1 for the blow-up of vorticity on the singular manifold, ω=O(s− β) in this case of vanishing angle ϕ.

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field ( m , n ) -closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.

Linear problems and hierarchies of Painlevé equations

In this paper, we show that the expansion of linear problems of the Painlevé equation in powers of the spectral variable can be used to derive hierarchies of ordinary differential equations. We applied this approach to linear problems of the first, second, third and fourth Painlevé equations. We derived a new hierarchy of the third Painlevé equation and rederived known hierarchies of the other equations. Moreover some special solutions of the hierarchies of the second, third and fourth Painlevé equations are also given.

Calculation of shocks using solutions of systems of ordinary differential equations

The method of intrinsic characterisation of shock wave propagation avoids the cumbersome task of solving the basic systems of equations before and after the shock, and has been used by various authors for direct calculation of relevant quantities on the shock. It leads to an infinite hierarchy of ordinary differential equations, which, due to the absence of a mathematical theory, is truncated to a finite system. In most practical cases, but not in all, the solutions of the truncated systems approximate the solution of the infinite system satisfactorily. The mathematical question of the error generated is completely open. We precisely define the concept of approximation and rigorously justify the local correctness of the approximation method for positive real analytic initial data for the inviscid Burgers’ equation, which has certain features in common with systems appearing in literature. At the same time we show that the nonuniqueness of the infinite system can lead to wrong results when the initial data are only C ∞ C^{\infty } and that blow-up of the solutions of the truncated systems are an obstacle for straightforward global approximation. Global approximation is achieved by recomputing the initial conditions for the approximating solutions in finitely many time steps. The results obtained will have to be taken into account in a future theory for more advanced systems.

Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations

We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate the limiting correlation functions to a nonlinear hierarchy of ordinary differential equations.

One generalization of the second Painlevé hierarchy

New integrable hierarchy of ordinary differential equations is presented. This hierarchy is shown to generalize the P2 hierarchy, while on the other hand, this hierarchy gives a special case of the third Painlevé equation. The isomonodromic linear problem is used to find this new hierarchy. The Painlevé approach to investigate the fourth-order ordinary differential equation of this hierarchy is applied.

On a Generalized 2 + 1 Dispersive Water Wave Hierarchy

We present a generalized non-isospectral dispersive water wave hierarchy in 2+1 dimensions. We characterize our entire hierarchy and its underlying linear problem using a single equation together with its corresponding non-isospectral scattering problem. This then allows a straightforward construction of linear problems for the entire generalized 2 + 1 hierarchy. Reductions of this hierarchy then yield new integrable hierarchies in 1+1 dimensions, and also new integrable hierarchies of ordinary differential equations, all together with their underlying linear problems. In particular, we obtain a generalized P_{IV} − P_{II} hierarchy; this includes as special cases both a hierarchy of ODEs having the fourth Painlevé equation as first member, and also a hierarchy of ODEs having the second Painlevé equation as first member. All of these hierarchies of ordinary differential equations, as well as their underlying linear problems, are new; both the P_{IV} hierarchy and the P_{II} hierarchy obtained here are different from those which have previously been given.

On a new non-isospectral variant of the Boussinesq hierarchy

We give a new non-isospectral extension to 2 + 1 dimensions of the Boussinesq hierarchy. Such a non-isospectral extension of the third-order scattering problem xxx +U x +(V - ) = 0 has not been considered previously. This extends our previous results on one-component hierarchies in 2 + 1 dimensions associated to third-order non-isospectral scattering problems. We characterize our entire (2 + 1)-dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non-isospectral scattering problem. This then allows an alternative approach to the construction of linear problems for the entire (2 + 1)-dimensional hierarchy. Reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems. In particular, we obtain a `fourth Painleve hierarchy', i.e. a hierarchy of ordinary differential equations having the fourth Painleveequation as its first member. We also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to Cosgrove.

Nonlinear fourth-order differential equations with solutions in the form of transcendents

We present two hierarchies of ordinary differential equations and give the relations between these hierarchies. We find rational and special solutions of one hierarchy. Solutions of two differential equations are shown to be essentially transcendental functions with respect to the integration constants.

Two hierarchies of ordinary differential equations and their properties

Two hierarchies of ordinary differential equations are presented. Relations between them are given. Rational and special solutions of one hierarchy are found.

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painlevé equation, . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of equations, and a little known Bäcklund transformation for itself. We then use our results on Bäcklund transformations to obtain, for each member of the hierarchy, a sequence of special integrals.

Moment hierarchies and cumulants in quantum optics.

Nonlinear problems in quantum optics can be described by an infinite hierarchy of ordinary differential equations for the moments. We discuss different truncation schemes involving cumulants. For illustration the method is applied to second-harmonic generation. We prove that all but second-order cumulants are equal for any quasiprobability distribution ${\mathit{P}}_{\mathit{s}}$, especially for the P, Q, and W functions.