278 research outputs found
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
Integrable viscous conservation laws
We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven.
We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe
Group analysis and renormgroup symmetries
An original regular approach to constructing special type symmetries for
boundary value problems, namely renormgroup symmetries, is presented. Different
methods of calculating these symmetries, based on modern group analysis are
described. Application of the approach to boundary value problems is
demonstrated with the help of a simple mathematical model.Comment: 17 pages, RevTeX LATeX file, to appear in Journal of Mathematical
Physic
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
Gurevich-Zybin system
We present three different linearizable extensions of the Gurevich-Zybin
system. Their general solutions are found by reciprocal transformations. In
this paper we rewrite the Gurevich-Zybin system as a Monge-Ampere equation. By
application of reciprocal transformation this equation is linearized.
Infinitely many local Hamiltonian structures, local Lagrangian representations,
local conservation laws and local commuting flows are found. Moreover, all
commuting flows can be written as Monge-Ampere equations similar to the
Gurevich-Zybin system. The Gurevich-Zybin system describes the formation of a
large scale structures in the Universe. The second harmonic wave generation is
known in nonlinear optics. In this paper we prove that the Gurevich-Zybin
system is equivalent to a degenerate case of the second harmonic generation.
Thus, the Gurevich-Zybin system is recognized as a degenerate first negative
flow of two-component Harry Dym hierarchy up to two Miura type transformations.
A reciprocal transformation between the Gurevich-Zybin system and degenerate
case of the second harmonic generation system is found. A new solution for the
second harmonic generation is presented in implicit form.Comment: Corrected typos and misprint
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