THE DAVEY-STEWARTSON MODEL IN QUADRATIC MEDIA: A WAY TO CONTROL PULSES
H. LEBLOND,
POMA., UMR CNRS 6136, Université d'Angers, 2 Bd Lavoisier 49045 Angers Cedex1, France.
1. A generalization of the NonLinear Schrödinger equation.
The (1+1)-dimensional theory of envelope soliton propagations rests on the nonlinear Schrödinger (NLS) equation, which reads:
iAuz+Buxx+Cu|u|2=0 (1)
x is the pulse shape variable, either transverse or longitudinal (time variable t). A well- known, but essential property of the NLS equation is to be completely integrable by means of the inverse scattering transform (IST) method for any value of the real coefficients A, B, and C. The most simple generalization of the NLS equation to (2+1) dimensions is the so-called two-dimensional nonlinear Schrödinger equation (2D NLS), that reads:
iAuz+Buxx+Cuyy+Du|u|2=0 (2)
Equation (2) is not integrable, whatever the value of the coefficients is, and does not admit any stable localized solution.
Another generalization of the NLS equation, that conserves many of its properties, is the Davey-Stewartson (DS) model, that reads:
iAuz+Buxx+Cuyy+Du|u|2+Euψ=0
αψxx+βψyy=γ|u|2xx+δ|u|2yy (3)
This model was first introduced to describe the two-dimensional evolution of water wave packets [1]. It involves an auxiliary field ψ, real-valued, that can be called rectified field, because in optics, it can result from optical rectification of the wave. This field describes a solitary wave, in the sense of a single isolated oscillation, and not of an envelope soliton (there is no carrier wave). The DS system reduces obviously to the 2DNLS equation if αδ−βγ=0 , but only in this case, and is thus more general.
For some values of its coefficients, the DS system (3) is one of the very few systems that are integrable by means of the IST method [2].It can always be reduced to:
iuz+ε1uxx+uyy+ε2u|u|2+ν ψu =0
ψxx+μψyy=|u|2xx (4) where and are real, and ε1,ε2=±1 . System (4) is integrable if:
μ=−ε1 and ν=−2ε2 (5)
Line solitons exist for all sign cases. Lumps, algebraically decaying in all directions, exist if ε1ε2=−1 [3]. Dromions, exponentially decaying in all directions, exist if ε1=+1 (DS I) [4]. The dromions involve nontrivial values for the auxiliary or rectified field at infinity. These values can describe the input of solitary waves, that interact with the main pulse.
2. Davey-Stewartson-type models in nonlinear optics.
The DS system (3) or (4) is a good generalization of the NLS equation (1) for two reasons: (i) Like the NLS equation, the DS system belongs to the set of the few partial differential equations that are completely integrable by means of the IST method. (ii) In the same way as, in (1+1) dimensions, multiscale expansions for envelope propagation almost always lead to the NLS equation, the model obtained through the same procedure in higher dimensional cases is most generally of DS type. Convergence of the asymptotic method from the mathematical point of view has been recently proved in a general frame [5]. A rigorous derivation of the model equations for soliton propagation in (2) materials of some crystal classes has been performed [6]. It involves three nonlinear mechanisms, listed in table 1. Far from phase-matching, the coefficients corresponding to the three mechanisms have the same order of magnitude. This is a priori true from the theoretical point of view of the multiscale expansion, and has been checked using published experimental values.
Table 1. nonlinear mechanisms.
Process Name Nonlinear coefficient
ω+ω−ω→ ω Third-order Kerr effect χ(3)(ω,ω,−ω,ω) ω+ω→ 2ω
2ω−ω→ ω
Cascading χ(2)(ω,ω,2ω)×χ(2)(2ω,−ω,2ω) ω−ω→ 0
0+ω→ω
Optical rectification
& electro-optic effect χ(2)(ω,−ω,0)×χ(2)(0,ω,ω)
Four polarizations, two for the fundamental wave and two for the rectified field, are taken into account. The derived model, that is always of DS-type, consists thus in four equations, apart from simplifications due to the symmetry [6]. It can be reduced to system (4) in some situations.
The two conditions under which these reduced DS type systems are completely integrable have been written down [7]. The first one expresses in a very simple way for the 6mm class:
1 ne2(0)
(
χ^xzx(2)(ω,−ω)
)
2+ 1ne2(2ω)
(
^χ8xzx(2) (ω,ω))
2=32 ^χ(3)xxxx(ω,ω,−ω) (6) It expresses an equilibrium between the coefficients of the third order Kerr effect and the sum of that of the quadratic processes. The second integrability condition relies the dispersion coefficient kk'' (where k() is the dispersion relation and k'' its second derivative) to the difference between the group velocity Vg of the wave and the velocity c/n(0) of the rectified field wave. The signs of the coefficients are so that lump exist (except in a special case), and that dromions would exist for anomalous dispersion.3. Solutions to the DS system : pulse control.
The DS-type systems (3) are classified into elliptic-elliptic, elliptic-hyperbolic, and so on, depending on the sign of the coefficients 1 and . The second equation of the DS system is hyperbolic if y is a the time variable t, and if the long wave travels slower than the main pulse u:
c
n(0)<vg(ω) (7)
Condition (7) is satisfied eg for KDP and for the lithium niobate. Then a solitary long wave, supported by the rectified field , is emitted by the main pulse (fig. 5).
Figure 5. The solitary wave emitted Figure 6. Geometry of the wave interaction.
by a two dimensional localized pulse.
In this case, the DS system describes an interaction between a localized envelope pulse u and two solitary long waves 1 and 2, of line-soliton shape, that is resonant in some special sense (fig. 6) [8]. Mathematically, 1 and 2 are the limit values at infinity of the rectified field . If u travels faster than the other waves, all will meet in the indicated interaction area. Efficiency of the interaction necessitates a velocity matching.
The waves are matched if the velocity of the intersection point of the two wave lines exactly coincides with that of the main pulse u. Then, seen from the main pulse, the two other waves apparently do not move, due to their self similarity along their line. The interaction lasts thus during all the propagation.
We present now some numerical solutions of the DS system (3), valid for electromagnetic wave propagation in ferromagnets [8]. Such behaviour can also arise in any physical situation for which the DS model is relevant. If the velocity matching is realized, through the interaction, it is possible to split the pulse into two parts (fig 8), or into four parts (fig. 9). In a physical situation where the self-phase modulation is defocusing, in the absence of incident matched solitary wave, the pulse will be spread out. The interaction is able to prevent this effect (fig. 10). When self-phase modulation is focusing, interaction can prevent collapse and stabilize the pulse (fig. 11).
Figure 8. The initial pulse has been split into Figure 9. The initial pulse has been split into 2 parts through the interaction. 4 parts through the interaction.
Figure 10. The interaction prevents the pulse. Figure 11. The interaction prevents collapse from spreading out. and stabilizes the pulse.
4. References :
1. A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A 338, 101-110 (1974).
2. A.S. Fokas and M.J. Ablowitz, Phys. Rev. Lett. 51 7-10 (1983).
3. J. Satsuma and M.J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys.
20 (7), 1496 (1979).
4. M. Boiti, J. J-P. Leon, L. Martina, and F. Pempinelli, Scattering of localized solitons in the plane, Phys.
Lett. A 132 (8-9), 432-439 (1988).
5. T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, preprint (1999).
6. H. Leblond, Propagation of optical localized pulses in (2) crystals :a (3+1)-dimensional model and its reduction to the NLS equation, J. Phys. A: Math. Gen. 31, 3041-3066 (1998).
7. H. Leblond, Bidimensional solitons in a quadratic medium, J. Phys. A: Math. Gen. 31, 5129-5143 (1998).
8. H. Leblond, Electromagnetic waves in ferromagnets: a Davey-Stewartson type model, J. Phys. A: Math.
Gen. 32, 7907-7932 (1999).
