8,943 research outputs found
Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method
In this work, we consider a family of nonlinear third-order evolution equations, where two
arbitrary functions depending on the dependent variable appear. Evolution equations of this type
model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to
cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling
waves solutions, all the the conservation laws that are invariant under translation symmetries are
directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave
ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution
equations under consideration into a first-order autonomous equation
N-wave interactions related to simple Lie algebras. Z_2- reductions and soliton solutions
The reductions of the integrable N-wave type equations solvable by the
inverse scattering method with the generalized Zakharov-Shabat systems L and
related to some simple Lie algebra g are analyzed. The Zakharov- Shabat
dressing method is extended to the case when g is an orthogonal algebra.
Several types of one soliton solutions of the corresponding N- wave equations
and their reductions are studied. We show that to each soliton solution one can
relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave
equations related to so(5) which find applications in Stockes-anti-Stockes wave
generation.Comment: 18 pages, 1 figure, LaTeX 2e, IOP-style; More clear exposition.
Introduction and Section 5 revised. Some typos are correcte
Nonholonomic Ricci Flows: II. Evolution Equations and Dynamics
This is the second paper in a series of works devoted to nonholonomic Ricci
flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows
of Riemannian metrics we can model mutual transforms of generalized
Finsler-Lagrange and Riemann geometries. We verify some assertions made in the
first partner paper and develop a formal scheme in which the geometric
constructions with Ricci flow evolution are elaborated for canonical nonlinear
and linear connection structures. This scheme is applied to a study of
Hamilton's Ricci flows on nonholonomic manifolds and related Einstein spaces
and Ricci solitons. The nonholonomic evolution equations are derived from
Perelman's functionals which are redefined in such a form that can be adapted
to the nonlinear connection structure. Next, the statistical analogy for
nonholonomic Ricci flows is formulated and the corresponding thermodynamical
expressions are found for compact configurations. Finally, we analyze two
physical applications: the nonholonomic Ricci flows associated to evolution
models for solitonic pp-wave solutions of Einstein equations, and compute the
Perelman's entropy for regular Lagrange and analogous gravitational systems.Comment: v2 41 pages, latex2e, 11pt, the variant accepted by J. Math. Phys.
with former section 2 eliminated, a new section 5 with applications in
gravity and geometric mechanics, and modified introduction, conclusion and
new reference
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