Quasi-linearization and stability analysis of some self-dual,dark equations and a new dynamical system?

Denis Blackmore,Mykola M Prytula and Anatolij K Prykarpatski

1 Department of Mathematical Sciences and CAMS,New Jersey Institute of Technology,Newark NJ 07102 United States of America

2 Ivan Franko National University of Lviv,Ukraine

3 Cracow University of Technology,Cracow,Poland

Abstract We describe a class of self-dual dark nonlinear dynamical systems a priori allowing their quasilinearization,whose integrability can be effectively studied by means of a geometrically based gradient-holonomic approach.A special case of the self-dual dynamical system,parametrically dependent on a functional variable is considered,and the related integrability condition is formulated.Using this integrability scheme,we study a new self-dual,dark nonlinear dynamical system on a smooth functional manifold,which models the interaction of atmospheric magnetosonic Alfvén plasma waves.We prove that this dynamical system possesses a Lax representation that allows its full direct linearization and compatible Poisson structures.Moreover,for this selfdual nonlinear dynamical system we construct an infinite hierarchy of mutually commuting conservation laws and prove its complete integrability.

Keywords:Hamiltonian system,Poisson structure,conservation laws,dark evolution system,asymptotic analysis,complete integrability

1.Introduction

Some twenty years ago,a new class of nonlinear dynamical systems,called ‘dark equations’ was introduced by Boris Kupershmidt[1,2],and shown to possess unusual properties that were not particularly well-understood at that time.Later,in related developments,some Burgers-type[3–5]and also Korteweg–de Vries type[6,7]dynamical systems were studied in detail,and it was proved that they have a finite number of conservation laws,a linearization and degenerate Lax representations,among other properties.In what follows,we provide a description of a class of self-dual dark-type(or just,dark,for short)nonlinear dynamical systems,which a priori allows their quasi-linearization,whose integrability can be effectively studied by means of a geometrically motivated[8,9–11]gradient-holonomic approach[12–14].Moreover,we study a slightly modified form of a self-dual nonlinear dark dynamical system on a functional manifold,whose integrability was recently analyzed in[15].Not only did this dynamical system appear to be Lax integrable,it also seemed to have a rich mathematical architecture including compatible Poisson structures and an infinite hierarchy of nontrivial mutually commuting conservation laws.In the sequel,we shall prove these properties using the gradient-holonomic integrability scheme devised in our prior work with several collaborators[12–14].

The remainder of this investigation is organized as follows.In section 2,we describe a rather wide class of self-dual dark quasi-linearized nonlinear dynamical systems on smooth functional manifolds.Then,in section 3,we study their quasilinearization property along with the integrability of a new selfdual nonlinear dark dynamical system.Next,in section 4,we prove the existence of a bi-Hamiltonian structure for the new system,and deduce its complete integrability.Finally,section 5 is devoted to summarizing our results and indicating some possible related future research directions.

2.Self-dual symmetry,quasi-linearization and nonlinear dark dynamical systems

We begin by studying integrability properties of a certain class of nonlinear dynamical systems of the form

3.Integrability description of dark equations

3.1.Integrability analysis

To proceed with the problem of classifying integrable dark dynamical systems on the functional manifold M,we need from the very beginning to analyze the conditions under which the reduced quasi-linearized system(2.3)possesses an infinite hierarchy of suitably ordered conservation laws.To do this,we will make use of the geometrically motivated gradient-holonomic integrability scheme devised in[16]and further developed in[12–14],to first transform the vector fields(2.3)to their following equivalent form on the functional manifold:

3.2.A new degenerate integrable dark type dynamical system

4.Bi-Hamiltonian structure and complete integrability

on the manifold Mu,whose solutions possess already much more nontrivial properties,and can suitably fit for modeling some physically interesting phenomena.

Remark 4.4.It is worth mentioning here that the self-dual dynamical system(3.17)as well as those(4.14)constructed above,are not contained within the usual Burgers type hierarchy,which can be easily checked by making use of the analysis done before in[4,7].

It is worth mentioning here that the integrability technique and results obtained above for the self-dual nonlinear dynamical system(3.17)on the functional manifold Mucan be effectively used via the corresponding Bogoyavlensky–Novikov reduction scheme[12–14,34]to describe a wide class of its both solitonic and finite-zoned quasiperiodic solutions in exact analytic form,which is now a topic of a work in progress.

5.Concluding remarks

We analyzed a new self-dual nonlinear dark dynamical system(on a functional manifold),which is a mathematical model of the interaction of atmospheric magneto-sonic Alfvén plasma waves.Using a gradient-holonomic approach,we were able to prove that the dynamical system is a completely integrable bi-Hamiltonian system,possessing an infinite hierarchy of nontrivial mutually commuting conservation laws with respect to the corresponding pair of compatible Poisson structures.It was noted that the gradient-holonomic approach,introduced by some of us and our collaborators,together with Bogoyavlensky–Novikov reduction,can apparently be used to construct a wide class of both solitonic and finite-zoned quasiperiodic solutions of the dark system in exact analytic form,and this is a goal that we are currently pursuing.

Acknowledgments

The authors are sincerely indebted to the Referees for their remarks and useful suggestions,which contributed to improving the manuscript.Especially AKP and MMP thank their colleagues D Dutych,Y Prykarpatsky and R Kycia for valuable discussions of computer-assisted calculational aspects related to the gradient-holonomic integrability approach.

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