The Nonlinear Schrödinger Equation And Its Possible Applications

J. Drozd * , Sreeram Valluri * , G. Papini + , and M. S. Sidharth o

*

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

o

Birla Science Centre, Hyderabad 500 463, India

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:

i

  

t

  2  

g

 2   0 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”.

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: 

t

  

x

 

xxx

 0 , 

const

.

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

m

2 )     2

p

 1    4

p

 1  0 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: 

p k

 

where k

 

A e



i k

  

x

  0

k

 ,     

k

 1  

k

  

A k

  2 4

m

0 2 2 

p



e p k

 2

ip k

  

x

2  1  

m

2 2  

A

4

m

2 4

p

 2

e p

 4

ip k

  

x

 1      2 1

p

and A is a const

.

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

H

       2 2

m

   2   1 2

g

       

d

 2

n

2  1   

where

  2 

k

 2  1

is



a



const

 2 .,   

we obtain

 0

the NLSE

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. Schrödinger’s equation for neutrons within the nucleus becomes the NLSE:

i

   

t

   2 2

m

  2  

g

      2  

V

 2  

i

  

t

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 

k

2

n

2    0

where k

2

n

2 

k

2 

U and k

2  2

mE

,

U

 2

mV k for the nonrelativ istic case

,

while

2 

E

2 

m

2 ,

U

 

V for the relativist ic case

.

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.