13,369 research outputs found
Refraction of dispersive shock waves
We study a dispersive counterpart of the classical gas dynamics problem of
the interaction of a shock wave with a counter-propagating simple rarefaction
wave often referred to as the shock wave refraction. The refraction of a
one-dimensional dispersive shock wave (DSW) due to its head-on collision with
the centred rarefaction wave (RW) is considered in the framework of defocusing
nonlinear Schr\"odinger (NLS) equation. For the integrable cubic nonlinearity
case we present a full asymptotic description of the DSW refraction by
constructing appropriate exact solutions of the Whitham modulation equations in
Riemann invariants. For the NLS equation with saturable nonlinearity, whose
modulation system does not possess Riemann invariants, we take advantage of the
recently developed method for the DSW description in non-integrable dispersive
systems to obtain main physical parameters of the DSW refraction. The key
features of the DSW-RW interaction predicted by our modulation theory analysis
are confirmed by direct numerical solutions of the full dispersive problem.Comment: 45 pages, 23 figures, minor revisio
Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity
For the hyperbolic system of quasilinear first-order partial differential
equations, linearizable by hodograph transformation, the conservation laws are
used to solve the Cauchy problem. The equivalence of the initial problem for
quasilinear system and the problem for conservation laws system permits to
construct the characteristic lines in domains, where Jacobian of hodograph
transformations is equal to zero. Moreover, the conservation laws give all
solutions of the linearized system. Some examples from the gas dynamics and
theory of plasticity are considered
Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability
A (d+1)-dimensional dispersionless PDE is said to be integrable if its
n-component hydrodynamic reductions are locally parametrized by (d-1)n
arbitrary functions of one variable. Given a PDE which does not pass the
integrability test, the method of hydrodynamic reductions allows one to
effectively reconstruct additional differential constraints which, when added
to the equation, make it an integrable system in fewer dimensions (if
consistent).Comment: 16 page
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
figure
Bogoliubov Renormalization Group and Symmetry of Solution in Mathematical Physics
Evolution of the concept known in the theoretical physics as the
Renormalization Group (RG) is presented. The corresponding symmetry, that has
been first introduced in QFT in mid-fifties, is a continuous symmetry of a
solution with respect to transformation involving parameters (e.g., of boundary
condition) specifying some particular solution.
After short detour into Wilson's discrete semi-group, we follow the expansion
of QFT RG and argue that the underlying transformation, being considered as a
reparameterisation one, is closely related to the self-similarity property. It
can be treated as its generalization, the Functional Self-similarity (FS).
Then, we review the essential progress during the last decade of the FS
concept in application to boundary value problem formulated in terms of
differential equations. A summary of a regular approach recently devised for
discovering the RG = FS symmetries with the help of the modern Lie group
analysis and some of its applications are given.
As a main physical illustration, we give application of new approach to
solution for a problem of self-focusing laser beam in a non-linear medium.Comment: Contribution to the proceedings of conference "RG 2000" (Taxco,
Mexico, Jan. 1999). To be published in Physics Report
Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate
We provide a classification of the possible flow of two-component
Bose-Einstein condensates evolving from initially discontinuous profiles. We
consider the situation where the dynamics can be reduced to the consideration
of a single polarization mode (also denoted as "magnetic excitation") obeying a
system of equations equivalent to the Landau-Lifshitz equation for an
easy-plane ferro-magnet. We present the full set of one-phase periodic
solutions. The corresponding Whitham modulation equations are obtained together
with formulas connecting their solutions with the Riemann invariants of the
modulation equations. The problem is not genuinely nonlinear, and this results
in a non-single-valued mapping of the solutions of the Whitham equations with
physical wave patterns as well as to the appearance of new elements --- contact
dispersive shock waves --- that are absent in more standard, genuinely
nonlinear situations. Our analytic results are confirmed by numerical
simulations
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