13,189 research outputs found
Periodic solutions for nonlinear equations with mean curvature-like operators
AbstractWe give an existence result for a periodic boundary value problem involving mean curvature-like operators in the scalar case. Following [R. Manásevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998), 367–393], we use an approach based on the Leray–Schauder degree
On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
In this article, we study the axisymmetric surface diffusion flow (ASD), a
fourth-order geometric evolution law. In particular, we prove that ASD
generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older
regular surfaces of revolution embedded in R^3 and satisfying periodic boundary
conditions. We also give conditions for global existence of solutions and prove
that solutions are real analytic in time and space. Further, we investigate the
geometric properties of solutions to ASD. Utilizing a connection to
axisymmetric surfaces with constant mean curvature, we characterize the
equilibria of ASD. Then, focusing on the family of cylinders, we establish
results regarding stability, instability and bifurcation behavior, with the
radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana
Isospectral deformations of the Dirac operator
We give more details about an integrable system in which the Dirac operator
D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a
Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) =
d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac
operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and
so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure
A survey of Hirota's difference equations
A review of selected topics in Hirota's bilinear difference equation (HBDE)
is given. This famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. Similarly to the continuous theory, HBDE is a member of an infinite
hierarchy. The central point of our exposition is a discrete version of the
zero curvature condition explicitly written in the form of discrete
Zakharov-Shabat equations for M-operators realized as difference or
pseudo-difference operators. A unified approach to various types of M-operators
and zero curvature representations is suggested. Different reductions of HBDE
to 2-dimensional equations are considered. Among them discrete counterparts of
the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical
examples are discussed in detail.Comment: LaTeX, 43 pages, LaTeX figures (with emlines2.sty
Periodic total variation flow of non-divergence type in Rn
We introduce a new notion of viscosity solutions for a class of very singular
nonlinear parabolic problems of non-divergence form in a periodic domain of
arbitrary dimension, whose diffusion on flat parts with zero slope is so strong
that it becomes a nonlocal quantity. The problems include the classical total
variation flow and a motion of a surface by a crystalline mean curvature. We
establish a comparison principle, the stability under approximation by
regularized parabolic problems, and an existence theorem for general continuous
initial data.Comment: 36 pages, 2 figure
Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
We study an equation lying `mid-way' between the periodic Hunter-Saxton and
Camassa-Holm equations, and which describes evolution of rotators in liquid
crystals with external magnetic field and self-interaction. We prove that it is
an Euler equation on the diffeomorphism group of the circle corresponding to a
natural right-invariant Sobolev metric. We show that the equation is
bihamiltonian and admits both cusped, as well as smooth, traveling-wave
solutions which are natural candidates for solitons. We also prove that it is
locally well-posed and establish results on the lifespan of its solutions.
Throughout the paper we argue that despite similarities to the KdV, CH and HS
equations, the new equation manifests several distinctive features that set it
apart from the other three.Comment: 30 pages, 2 figure
Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings
Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model
(GN2), and its chiral cousin, the NJL2 model, have shown that there are phases
with inhomogeneous crystalline condensates. These (static) condensates can be
found analytically because the relevant Hartree-Fock and gap equations can be
reduced to the nonlinear Schr\"odinger equation, whose deformations are
governed by the mKdV and AKNS integrable hierarchies, respectively. Recently,
Thies et al have shown that time-dependent Hartree-Fock solutions describing
baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation,
and can be mapped directly to classical string solutions in AdS3. Here we
propose a geometric perspective for this result, based on the generalized
Weierstrass spinor representation for the embedding of 2d surfaces into 3d
spaces, which explains why these well-known integrable systems underlie these
various Gross-Neveu gap equations, and why there should be a connection to
classical string theory solutions. This geometric viewpoint may be useful for
higher dimensional models, where the relevant integrable hierarchies include
the Davey-Stewartson and Novikov-Veselov systems.Comment: 27 pages, 1 figur
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