M ay 2 00 5 Resolution of a shock in hyperbolic systems modified by weak dispersion

نویسنده

  • G. A. El
چکیده

We present a way to deal with dispersion-dominated " shock-type " transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The analysis is performed by assuming that, the dispersive shock transition between two different constant states can be modelled by an expansion fan solution of the associated modulation (Whitham) system for the short-wavelength nonlinear oscillations in the transition region (the so-called Gurevich – Pitaevskii problem). We consider as single-wave so bi-directional systems. The main mathematical assumption is that of hyperbolicity of the Whitham system for the solutions of our interest. By using general properties of the Whitham averaging for a certain class of nonlinear dispersive systems and specific features of the Cauchy data prescription on characteristics we derive a set of transition conditions for the dispersive shock, actually bypassing full integration of the modulation equations. Along with model KdV and mKdV examples, we consider a non-integrable system describing fully nonlinear ion-acoustic waves in collisionless plasma. In all cases our transition conditions are in complete agreement with previous analytical and numerical results. Modern theory of dispersive shocks is based on the analysis of the Whitham averaged equations describing modulations of nonlinear short-wavelength oscillations in the transition region between two smooth regimes. If the wave dynamics is governed by one of the completely integrable equations, exact solutions in terms of the Riemann invariants are also available for the corresponding Whitham system providing full asymptotic description of such a transition. For nonintegrable systems describing many physically important cases of nonlinear dispersive wave propagation such modulation solutions are not readily (if at all) available. In this paper we show that, by using some general properties of the Whitham equations connected with their " averaged " origin one is able to obtain a set of transition conditions representing the " dispersive " analog of the 1 traditional shock conditions of classical dissipative gas dynamics. The developed method does not make use of the Riemann invariants for the modulation equations and can be applied to nonintegrable conservative systems. In particular, the obtained conditions allow for determination of the lead solitary wave amplitude in terms of the jumps for hydrodynamic variables across the dispersive shock.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Resolution of a shock in hyperbolic systems modified by weak dispersion.

We present a way to deal with dispersion-dominated "shock-type" transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The analysis is performed by assuming that the dispersive shock transition between two different constant states can be modeled by an expansion fan solution of th...

متن کامل

06 Resolution of a shock in hyperbolic systems modified by weak dispersion

We present a way to deal with dispersion-dominated " shock-type " transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The analysis is performed by assuming that, the dispersive shock transition between two different constant states can be modelled by an expansion fan solution o...

متن کامل

2 4 M ay 2 00 5 Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CH, n ≥ 3. For the quaternionic hyperbolic spaces HH, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificatio...

متن کامل

ar X iv : h ep - t h / 03 05 03 7 v 1 5 M ay 2 00 3 FORMS ON VECTOR BUNDLES OVER COMPACT REAL HYPERBOLIC MANIFOLDS

We study gauge theories based on abelian p− forms on real compact hyperbolic manifolds. The tensor kernel trace formula and the spectral functions associated with free generalized gauge fields are analyzed.

متن کامل

ar X iv : m at h / 03 04 27 8 v 2 [ m at h . G R ] 1 5 M ay 2 00 3 IDEAL BICOMBINGS FOR HYPERBOLIC GROUPSAND APPLICATIONS

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [MSb] hold for all non-elementary hyperbolic groups and thei...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2005