Spatial solitons

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In optics spatial solitons are the particle-like inhomogeneties 
of electromagnetic  field. The most known example is a soliton of 
Nonlinear Shrodinger Equation (NLS):

			(1)

describing filaments occuring when light beam passes through 
Kerr-like medium: dielectric, semiconductor, liquid, gas or even 
vacuum. You can see below, how smooth laser beam coming from the
left passes through the Kerr medium. Bright spots in transverse 
section appear after the passage, called filaments.   


Equation (1) is not easy to solve. Some authors found elegant
solutions. Among the most known are Bespalov and Talanov's result,
who found relation between width of filaments and speed of their growth. 
According to their theory filaments originate from noisy harmonics:

			(2)

The first exponent is plane wave solution of Eq.1, while 

is noisy modulation which leads to formation of the filaments.The 
speed of the filaments growth, i.e.increments are given by:

			(3)

where  is inverse size of filament.The positive root   
shows range of instability:

Propagation solitons

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Noteworthy, the above solution is valid for small modulations, thus for
linear stage of instability. Solitons are produced at nonlinear stage
of instability.For one transverse dimension the equation: 

				(6)

had been solved by V.E.Zakharov and A.B.Shabat exactly.
Among the solutions obtained the simplest is a single soliton:

 


The soliton propagates without changing its shape until it find 
inhomegeneity or another soliton.

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Solitons in bistable cavity

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The existence of solitons is due to balance of diffraction and nonlinearity,
i.e. positive lens induced in Kerr medium by radiation:



In dissipative systems this balance is accompanied by balance of gain 
and losses.In optics examples are bistable cavity with external 
light injection: 



and in bistable laser, i.e. laser with saturable absorber inside.
Intracavity field exceeds incoming field and shows histeresis:

	

The above figure shows phenomenon of bistability: intracavity field
is multiple-value function of incoming field. The switching between
lower and upper branch of histeresis curve produces solitary waves,
which is referred to as autosolitons(See N.N.Rosanov).

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Solitons in confocal cavities

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Succesfully I found exact solution in for spatial soliton for a 
specific case of bistable cavity.It is a cavity with a spherical 
mirrors, whose foci are placed in the same point. Such confocal 
microcavity contains thin gain and absorber slices, placed in 
Fourier-conjugate planes, i.e. directly onto mirrors:

	

The solution obtained has exact form of sech soliton for both
absorbing mirror(left): 

	(7)

and amplifying mirror(right) connected by Fourier transform:

	

The solution hed been obtained by discrete map approach,
by means of asymptotic evaluation of nonlinear integral equation:



Notatons and definitions are in our recent paper.	

The above solution predicted akin spatial solitons observed in confocal 
cavity with saturable absorber and in numerical experiment 
performed by V.B.Taranenko,K.Staliunas,C.O.Weiss:



The exact solution (7) gives, for example, correct description
 of soliton width near lasing threshold:



 
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