53,030 research outputs found
Nonlinear deterministic equations in biological evolution
We review models of biological evolution in which the population frequency
changes deterministically with time. If the population is self-replicating,
although the equations for simple prototypes can be linearised, nonlinear
equations arise in many complex situations. For sexual populations, even in the
simplest setting, the equations are necessarily nonlinear due to the mixing of
the parental genetic material. The solutions of such nonlinear equations
display interesting features such as multiple equilibria and phase transitions.
We mainly discuss those models for which an analytical understanding of such
nonlinear equations is available.Comment: Invited review for J. Nonlin. Math. Phy
Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations
We obtain new gauge-invariant forms of two-dimensional integrable systems of
nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the
generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov
system. We show how these forms imply both new and well-known two-dimensional
integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt
equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and
modified Nizhnik-Veselov-Novikov equation. We consider Miura-type
transformations between nonlinear equations in different gauges.Comment: Talk given at the Workshop "Nonlinear Physics: Theory and Experiment.
V", Gallipoli (Lecce, Italy), 12-21 June, 200
Linear Superposition in Nonlinear Equations
Even though the KdV and modified KdV equations are nonlinear, we show that
suitable linear combinations of known periodic solutions involving Jacobi
elliptic functions yield a large class of additional solutions. This procedure
works by virtue of some remarkable new identities satisfied by the elliptic
functions.Comment: 7 pages, 1 figur
Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations
For a large number of nonlinear equations, both discrete and continuum, we
demonstrate a kind of linear superposition. We show that whenever a nonlinear
equation admits solutions in terms of both Jacobi elliptic functions \cn(x,m)
and \dn(x,m) with modulus , then it also admits solutions in terms of
their sum as well as difference. We have checked this in the case of several
nonlinear equations such as the nonlinear Schr\"odinger equation, MKdV, a mixed
KdV-MKdV system, a mixed quadratic-cubic nonlinear Schr\"odinger equation, the
Ablowitz-Ladik equation, the saturable nonlinear Schr\"odinger equation,
, the discrete MKdV as well as for several coupled field
equations. Further, for a large number of nonlinear equations, we show that
whenever a nonlinear equation admits a periodic solution in terms of
\dn^2(x,m), it also admits solutions in terms of \dn^2(x,m) \pm \sqrt{m}
\cn(x,m) \dn(x,m), even though \cn(x,m) \dn(x,m) is not a solution of these
nonlinear equations. Finally, we also obtain superposed solutions of various
forms for several coupled nonlinear equations.Comment: 40 pages, no figure
Nonclassical symmetries as special solutions of heir-equations
In (Nucci M.C. 1994, Physica D 78 p.124), we have found that iterations of
the nonclassical symmetries method give rise to new nonlinear equations, which
inherit the Lie point symmetry algebra of the given equation. In the present
paper, we show that special solutions of the right-order heir-equation
correspond to classical and nonclassical symmetries of the original equations.
An infinite number of nonlinear equations which possess nonclassical symmetries
are derived
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