43,907 research outputs found
Supersymmetric methods in the traveling variable: inside neurons and at the brain scale
We apply the mathematical technique of factorization of differential
operators to two different problems. First we review our results related to the
supersymmetry of the Montroll kinks moving onto the microtubule walls as well
as mentioning the sine-Gordon model for the microtubule nonlinear excitations.
Second, we find analytic expressions for a class of one-parameter solutions of
a sort of diffusion equation of Bessel type that is obtained by supersymmetry
from the homogeneous form of a simple damped wave equations derived in the
works of P.A. Robinson and collaborators for the corticothalamic system. We
also present a possible interpretation of the diffusion equation in the brain
contextComment: 14 pages, 1 figur
Solutions of the Perturbed KDV Equation for Convecting Fluids by Factorizations
In this paper, we obtain some new explicit travelling wave solutions of the
perturbed KdV equation through recent factorization techniques that can be
performed when the coefficients of the equation fulfill a certain condition.
The solutions are obtained by using a two-step factorization procedure through
which the perturbed KdV equation is reduced to a nonlinear second order
differential equation, and to some Bernoulli and Abel type differential
equations whose solutions are expressed in terms of the exponential and
Weierstrass functionsComment: 4 pages, some changes in the text according to referees' suggestions,
added one reference, accepted at Central Europ. J. Phy
Nonlinear second order ODE's: Factorizations and particular solutions
We present particular solutions for the following important nonlinear second
order differential equations: modified Emden, generalized Lienard, convective
Fisher, and generalized Burgers-Huxley. For the latter two equations these
solutions are obtained in the travelling frame. All these particular solutions
are the result of extending a simple and efficient factorization method that we
developed in Phys. Rev. E 71 (2005) 046607Comment: 6 pages, v3=published versio
Traveling wave solutions for wave equations with two exponential nonlinearities
We use a simple method that leads to the integrals involved in obtaining the
traveling wave solutions of wave equations with one and two exponential
nonlinearities. When the constant term in the integrand is zero, implicit
solutions in terms of hypergeometric functions are obtained while when that
term is nonzero all the basic traveling wave solutions of Liouville, Tzitzeica
and their variants, as well as sine/sinh-Gordon equations with important
applications in the phenomenology of nonlinear physics and dynamical systems
are found through a detailed study of the corresponding elliptic equationsComment: 9 pages, 7 figures, 42 references, version matching the published
articl
Riccati nonhermiticity with application to the Morse potential
A supersymmetric one-dimensional matrix procedure similar to relationships of
the same type between Dirac and Schrodinger equations in particle physics is
described at the general level. By this means we are able to introduce a
nonhermitic Hamiltonian having the imaginary part proportional to the solution
of a Riccati equation of the Witten type. The procedure is applied to the
exactly solvable Morse potential introducing in this way the corresponding
nonhermitic Morse problem. A possible application is to molecular diffraction
in evanescent waves over nanostructured surfacesComment: 8 pages, 4 figure
Classical harmonic oscillator with Dirac-like parameters and possible applications
We obtain a class of parametric oscillation modes that we call K-modes with
damping and absorption that are connected to the classical harmonic oscillator
modes through the "supersymmetric" one-dimensional matrix procedure similar to
relationships of the same type between Dirac and Schroedinger equations in
particle physics. When a single coupling parameter, denoted by K, is used, it
characterizes both the damping and the dissipative features of these modes.
Generalizations to several K parameters are also possible and lead to
analytical results. If the problem is passed to the physical optics (and/or
acoustics) context by switching from the oscillator equation to the
corresponding Helmholtz equation, one may hope to detect the K-modes as
waveguide modes of specially designed waveguides and/or cavitiesComment: 14 pages, 9 figures, revised, accepted at J. Phys.
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