We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.
Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symple…
We classify the admissible transformations in a class of variable coefficient Korteweg--de Vries equations. As a result, full description of the structure of the equivalence groupoid of the class is given. The class under study is partitioned into six disjoint normalized subclasses. The widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases. Ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes. The group classification of one of the subclas…
Preliminary results about Lie and potential symmetries of a class of Korteweg-de Vries type equations are presented. In order to prove existence of potential symmetries three different systems of so called determining equations are analysed. It is shown that two systems lead only to Lie symmetries while the third system produces potential symmetries provided the equation in question has an appropriate structure
We consider the modified Korteweg-de Vries equation, $ u_{xxx} + 6u^2 u_x + u_t \ =\ 0 $, and explore its dynamics in {\it spatial} direction. Higher $x$ derivatives bring about the {\it ghosts}. We argue that these ghosts are benign, i.e. the classical dynamics of this system does not involve a blow-up. This probably means that also the associated quantum problem is well defined.
In this paper we discuss some properties of linear fractional dispersive waves. In particular, we compare the dispersion relations emerging from the kinematic wave equation and from the linearised Korteweg - de Vries equation with the corresponding time-fractionalized versions. For this purpose, we evaluate the expressions for the phase velocity and for the group velocity, highlighting the differences not only analytically, but also by means of illuminating plots.
We establish the nonlinear stability of solitary waves (solitons) and periodic traveling wave solutions (cnoidal waves) for a Korteweg-de Vries (KdV) equation which includes a fifth order dispersive term. The traveling wave solutions which yield solitons for zero boundary conditions and wave-trains of cnoidal waves for nonzero boundary conditions are analyzed using stability theorems, which rely on the positivity properties of the Fourier transforms. We show that all families of solutions considered here are (orbitally) stable.
Following Deift-Zhou's nonlinear steepest descent method, the long-time asymptotic behavior for the Cauchy problem of the 5th order modified Korteweg-de Vries equation is analyzed. Based on the inverse scattering transform, the 5th order MKdV is transformed to a 2 by 2 oscillatory Riemann-Hilbert problem, then by manipulating the Cauchy operator and reducing the degree of the phase function, the long-time asymptotics of the solution is given in terms of solutions of the parabolic cylinder equation.
We consider the linearized Korteweg-de-Vries equa- tions, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that pro- perties of the induced dynamics can be obtained by studying boundary operators in the corresponding boundary space indu- ced by the vertices of the graph. In particular, we characterize unitary dynamics and contractive dynamics. We demonstrate our results on various special graphs, including those recently treated in the literature.
We present in this work the hamiltonian formulation of an octonionic extension for the Korteweg-de Vries equation. The formulation takes into account the non commmutativity and non associativity of the implicit algebra which defines the equation. We also analize the Poisson structure of the hamiltonian formulation. We propose a parametric master Lagrangian which contains the two hamiltonian structures of the integrable octonionic equation.
We address the problem of long-time asymptotics for the solutions of the Korteweg-de Vries equation under low regularity assumptions. We consider rapidly decreasing initial data admitting only a finite number of moments. For the so-called "soliton region", an improved asymptotic estimate is provided, in comparison with the one already present in the literature. Our analysis is based on the dbar steepest descent method proposed by P. Miller and K. T. D. -R. McLaughlin.
We describe the equivalence groupoid of the class of general Burgers - Korteweg - de Vries equations with space-dependent coefficients. This class is shown to reduce by a family of equivalence transformations to a subclass whose usual equivalence group is four-dimensional. Classified are admissible transformations of this subclass and singled out are its subclasses admitting maximal nontrivial conditional equivalence groups. All of them turn out to have dimension higher than four. In particular, a few new examples of nontrivial cases of normalization in the generalized sense of classes of diff…
In the paper, we consider a stochastic hybrid Korteweg - de Vries - Burgers type equation with multiplicative noise in the form of cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. The proof of the existence of the solution is based on two approximations of the considered problem and the compactness method. First, we introduce an auxiliary problem corresponding to the equation studied. Then, we prove the existence of a martingale solution to this problem. Finally, we show that the solution of the auxiliary problem converges, in some sense, to …
We propose and demonstrate, with the one-dimensional Korteweg-de Vries-Burgers model, the scenarios of transfer loop and \textit{all-scale} statistical equilibrium, the former being associated to shock formation and the latter to Gaussian distributions as in a canonical ensemble, but with wavelength-dependent temperatures. The discussions emphasize, among the multi-disciplinary relevance, the classical nonlinear acoustics and quantum shock waves, for the possibility of more favorable experimental tests.
