The Nonlinear Schrödinger Equation And Its Possible ApplicationsJ. 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 0A2oBirla Science Centre, Hyderabad 500 463, India

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 stopped—not 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:

. , 0const xxx x t

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:

022

gt

i

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”.

Solitary wave solutions of nonlinear field equations have been studied in several areas of physics. Field equations considered have the form:

0)( 14122 ppm

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:

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:

dgm

H

2

1

22

2

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:

gmt

i 22

2

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:

.

,

,

2,2

0

222

2

222

222

caseicrelativistthefor

xVxUmEk

whilecaseisticnonrelativthefor

xmVxUmEkand

xUknkwhere

xnk

Papini more specifically assumed that the refractive index is proportional to the density of incoming particles. Then by setting:

01

.,

1

222

22

xk

NLSEtheobtainweconstaiswhere

n

The nonlinear phenomena exhibited by nuclei when they are probed by intense particle beams is a topic in nonlinear nuclear dynamics meriting serious study. Schrödinger’s equation for neutrons within the nucleus becomes the NLSE:

tiV

22

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.

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.

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