The Nonlinear Schr

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The Nonlinear Schrödinger Equation And Its
Possible Applications J. Drozd, Sreeram
Valluri, G. Papini, and M. S.
Sidhartho Department of Applied Mathematics and
Department of Physics Astronomy, University of
Western Ontario, London, Ontario N6A
5B7 Department of Physics Astronomy,
University of Regina, Regina, Saskatchewan S4S
0A2 oBirla Science Centre, Hyderabad 500 463,
India
The NLSE also appears in the description of a
Bose-Einstein condensate, a context where it is
often called the Gross-Pitaevskii equation. It
admits solutions in the form of coherent
structures like vortices that define states that
can be excited in superfuild helium. Consider a
Bose gas with weak pair repulsions between atoms.
The Hamiltonian of such a system in the second
quantization representation has the form
The name Nonlinear Schrödinger equation (NLSE)
originates from a formal analogy with the
Schrödinger equation of quantum mechanics. In
this context a nonlinear potential arises in the
mean field description of interacting
particles. The elliptic NLSE, which when
written in a frame moving at the group velocity
of the carrying wave takes the simple form
Papini more specifically assumed that the
refractive index is proportional to the density
of incoming particles. Then by setting
with an attractive (g 1) or repulsive (g 1)
nonlinearity with its generalization to arbitrary
power-law nonlinearities g ? 2? ?. The NLSE
gives what are termed as solitary wave
solutions.
This Hamiltonian, when differentiated with
respect to ?, corresponds to the equation of
motion for the Heisenberg operator ?, and reduces
to the Gross-Pitaevskii or NLSE
The nonlinear phenomena exhibited by nuclei when
they are probed by intense particle beams is a
topic in nonlinear nuclear dynamics meriting
serious study. Schrödingers equation for
neutrons within the nucleus becomes the NLSE
The concept of a solitary wave was introduced to
the budding science of hydrodynamics well over a
century ago by Scott-Russell with the following
delightful description I was observing the
motion of a boat which was rapidly drawn along a
narrow channel by a pair of horses, when the boat
suddenly stoppednot so the mass of water in the
channel which it had put in motion it
accumulated round the prow of the vessel in a
state of violent agitation, then suddenly leaving
it behind, rolled forward with great velocity,
assuming the form of a large solitary elevation,
a rounded, smooth and well-defined heap of water,
which continued its course along the channel
apparently without change of form or diminution
of speed. I followed it on horseback, and
overtook it still rolling on at a rate of some
eight or nine miles an hour, preserving its
original figure some thirty feet long and a foot
to a foot and a half in height. Its height
gradually diminished, and after a chase of one or
two miles I lost it in the windings of the
channel. Such, in the month of August 1834, was
my first chance interview with that singular and
beautiful phenomenon. . . . In 1895 Korteweg
and deVries provided a simple analytic foundation
for the study of solitary waves by developing an
equation for shallow water waves which includes
both linear and dispersive effects but ignores
dissipation
Generalizing the Gross-Pitaevskii equation with
nonzero ? and ? terms of opposite sign, one can
get higher nonlinearities which can help explain
such physical phenomena such as non-adiabatic
spin flips that can create effects which suppress
the condensation. One application of
nonlinearity is solving the NLSE for a
double-well potential as described by Razavy.
This interesting bistable potential is the case
for the equation of motion describing the normal
modes of vibration of a stretched membrane of
variable density. Double-well potentials have
been used in the quantum theory of molecules to
describe the motion of a particle in the presence
of two centres of force, such as in the Morse
Potential. In the theory of optical-potential
scattering, Papini treats nuclear matter as an
optical medium of index of refraction n. The
Schrödinger and Klein-Gordon equations can in
fact be recast in the form
Aközbek and John have analyzed finite energy
solitary waves in two- and three-dimensional
periodic structures exhibiting a complete
photonic band gap in terms of an effective
nonlinear Dirac equation. Using a linear
stability analysis, they derive a criterion for
the stability of solitary wave solutions of the
nonlinear Dirac equation analogous to the
criterion for the NLSE.
Solitary wave solutions of nonlinear field
equations have been studied in several areas of
physics. Field equations considered have the
form
References G. Papini, Nuclear Matter as a
Nonlinear Optical Medium, Lettere al Nuovo
Cimento, Vol. 17, No. 12, Nov. 1976, pp.
419-420. M. Razavy, An exactly soluble
Schrödinger equation with a bistable potential,
Am. J. Phys., Vol. 48, No. 4, Apr. 1980, pp.
285-288. F. Dalfovo, S. Giorgini, L. P.
Pitaevskii, S. Stringari, Theory of
Bose-Einstein condesation in trapped gases,
Reviews of Modern Physics, Vol. 71, No. 3, Apr.
1999, pp. 463-512. N. Aközbek, S. John, Optical
solitary waves in two- and three-dimensional
nonlinear photonic band-gap structures, Physical
Review E, Vol. 57, No. 2, Feb. 1998, pp.
2287-2319. J. Drozd, G. Papini, M. Sidharth, S.
R. Valluri, A Problem in Non-linear Nuclear
Dynamics (unpublished). P. B. Burt, Solitary
Waves in Nonlinear Field Theories, Physical
Review Letters, Vol. 32, No. 19, May 1974, pp.
1080-1081. P. B. Burt, J. L. Reid, Exact
Solution to a Nonlinear Klein-Gordon Equation,
Journal of Mathematical Analysis and
Applications, Vol. 55, 1976, pp.
43-45. Catherine Sulem, Pierre-Louis Sulem, The
Nonlinear Schrödinger Equation Self-Focusing and
Wave Collapse, Applied Mathematical Sciences,
Volume 139, Springer, New York, 1999.
with p ? 0, 1/2, 1. The solutions reduce to
positive or negative frequency plane-wave
solutions of the Klein-Gordon equation in the
limit of vanishing coupling constants. With p
1, ? ? 0, and ? 0, this equation reduces to the
NLSE. The solitary wave solution of the
differential equation is given below