Burgers and Kadomtsev-Petviashvili hierarchies: A functional representation approach

Functional representations of (matrix) Burgers and potential Kadomtsev-Petviashvili (pKP) hierarchies (and others), as well as some corresponding Bäcklund transformations, can be obtained surprisingly simply from a “discrete” functional zero-curvature equation. We use these representations to show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the solutions of certain functional Riccati equations.     

Integrable equations on ℤ-graded Lie algebrasLie algebras

We consider evolution systems admitting L-A-pairs in ℤ-graded Lie algebras. We relate several hierarchies of integrable systems to a single L operator. The different hierarchies corresponds to different decompositions of the zero component of a ℤ-graded algebra into the sum of two subalgebras. This allows us to construct new examples of multi-component integrable system following the Burgers, mKdV, NLS and Boussinesq equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 3, pp. 375–383 September, 1997.     

The method of generalized Cole-Hopf substitutions and new examples of linearizable nonlinear evolution equations

We propose a new approach for constructing nonlinear evolution equations in matrix form that are integrable via substitutions similar to the Cole-Hopf substitution linearizing the Burgers equation. We use this new approach to find new integrable nonlinear evolution equations and their hierarchies. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 58–71, January, 2009     

Asymptotic Reynolds-averaged closure of nonlinear ensembles

The authors show that the Reynolds-averaged representation offinite-dimensional dynamical systems with quadratic nonlinearitycan be closed in the limit of infinite moment. The closure isposed for problems with algebraic moment hierarchies, and testedon regular ensembles of the viscous Burgers equation and ensembleswhose individual members are chaotic solutions of the Lorenzequations. The solvability of the averaged system of equationsand the equivalence of direct simulation and Reynolds-averagedcomputation, using a simple one-dimensional model problem, aredemonstrated.     

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier–Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (\(1+1\))-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

     

A Hamiltonian Regularization of the Burgers Equation

We consider a quasilinear equation that consists of the inviscid Burgers equation plus O(α²) nonlinear terms. As we show, these extra terms regularize the Burgers equation in the following sense: for smooth initial data, the α > 0 equation has classical solutions globally in time. Furthermore, in the zero-α limit, solutions of the regularized equation converge strongly to weak solutions of the Burgers equation. We present numerical evidence that the zero-α limit satisfies the Oleinik entropy inequality. For all α ≥ 0, the regularized equation possesses a nonlocal Poisson structure. We prove the Jacobi identity for this generalized Hamiltonian structure.     

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.     

Initial Boundary Value Problems for Scalar and Vector Burgers Equations

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x >0, t >0 with Riemann type of initial and boundary conditions and use the Hopf–Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.     

Construction of a hierarchy of negative‐order integrable Burgers equations of higher orders

In this work, we employ the recursion operator, the Burgers equation and its inverse operator, for constructing a hierarchy of negative‐order integrable Burgers equations of higher orders. The complete integrability of each established equation emerges by virtue of the correlation between integrability and recursion operators. We use the simplified Hirota's method to obtain multiple kink solutions for some of the derived equations, and in particular, for the generalized negative‐order Burgers equation.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author, at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by Golse and Perthame.     

Hierarchies of exact solutions for the discrete third Painlevé equation

In this paper, we construct hierarchies of rational solutions of the discrete third Painlevé equation (d-PIII) by applying Schlesinger transformations to simple initial solutions. We show how these solutions reduce in the continuous limit to the hierarchies of rational solutions of the third Painlevé equation (PIII). We also study the solutions of d-PIII which are expressed in terms of discrete Bessel functions and show that these solutions reduce in the continuous limit the hierarchies of special function solutions of PIII.     

Asymptotic N-wave Solutions of the Nonplanar Burgers Equation

In this paper, we construct asymptotic N-wave solutions for the nonplanar Burgers equation as   t →∞  via a balancing argument. These constructed asymptotics are compared with the approximate solutions of the nonplanar Burgers equation obtained by an approach due to Parker ( Acoust. Lett. 4 (1981)). We also present a computationally convenient form for the N-wave solution of the nonplanar Burgers equation modifying Sachdev et al.'s ( Stud. Appl. Math . 103 (1999)) approach. The asymptotic N-wave solutions obtained by balancing argument and modification to Sachdev et al.'s approach are validated by a careful numerical study.     

A stochastic variational approach to viscous Burgers equations

We consider viscous Burgers equations in one dimension of space and derive their solutions from stochastic variational principles on the corresponding group of homeomorphisms. The metrics considered on this group are L ^p metrics. The velocity corresponds to the drift of some stochastic Lagrangian processes. Existence of minima is proved in some cases by direct methods. We also give a representation of the solutions of viscous Burgers equations in terms of stochastic forward-backward systems.     

Towards a new theory of confirmation

Any sequence of events can be “explained” by any of an infinite number of hypotheses. Popper describes the “logic of discovery” as a process of choosing from a hierarchy of hypotheses the first hypothesis which is not at variance with the observed facts. Blum and Blum formalized these hierarchies of hypotheses as hierarchies of infinite binary sequences and imposed on them certain decidability conditions. In this paper we also consider hierarchies of infinite binary sequences but we impose only the most elementary Bayesian considerations. We use the structure of such hierarchies to define “confirmation”. We then suggest a definition of probability based on the amount of confirmation a particular hypothesis (i.e. pattern) has received. We show that hypothesis confirmation alone is a sound basis for determining probabilities and in particular that Carnap’s logical and empirical criteria for determining probabilities are consequences of the confirmation criterion in appropriate limiting cases.     

Travelling waves for a non-local Korteweg–de Vries–Burgers equation

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.     

Shock waves for the Burgers equation and curvatures of diffeomorphism groups

We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein geometry of the space of densities.     

Block-diagonal semidefinite programming hierarchies for 0/1 programming

Lovász and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for 0/1 linear programming problems. We revisit these two constructions and propose two new, block-diagonal hierarchies, which are at least as strong as the Lovász-Schrijver hierarchy, but less costly to compute. We report experimental results for the stable set problem of Paley graphs.     

Discrete Burgers’ Equation, Binomial Coefficients and Mandala

In discrete Burgers’ equation, time is discretized but space remains continuous. Reduction modulo 2 of discrete Burgers’ equation discretizes also space. This dynamical system generates an elegant pattern known since long time ago. We also propose to show that the Galois groups of particular solutions of Burgers’ equation are abelian. It would imply that the dynamical systems are integrable.     

Blow-up and regularity problems of hypoviscous fluid equations

We consider 1D Burgers equation with hypoviscosity at a critical exponent under periodic boundary conditions. Assuming global (in time) regularity, we derive an asymptotic equation by a simple transformation. It turned out that it has exactly the same form as the one obtained by applying Moore's asymptotic analysis, originally developed for the Birkhoff-Rott equation, to the inviscid Burgers equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)