Interaction between positive and negative energy modes in the presence of second order dispersive effects

PHYSICS LETTERS

Volume 48A, number 1

20 May 1914

INTERACTION BETWEEN POSITIVE AND NEGATIVE ENERGY MODES IN THE PRESENCE OF SECOND ORDER DJSPERSIVE EFFECTS J. ASKNE Research Laboratov

of Electronics,

Chalmers University of Technology,

Giiteborg, Sweden

K.B. DYSTHE and O.T. GUDMESTAD* 7he University of TromsG, The Aurora1 Observatory Tromsii, Norway

J. WEILAND** Institute for Electromagnetic

and H. WILHELMSSON

Field Theory, Chalmers University of Technology,

GBteborg, Sweden

Received 1 April 1974 The influence of second order dispersive effects on a system of three interacting waves, stabilized by nonlinear frequency shifts, is studied in a coherent wave description. A strong tendency for a new type of rapid oscillations in amplitudes and phases of the waves has been discovered. The growth of such oscillations may even drastically change the evolution of the system. Besides the second order dispersive effects may destabilize the system by counteracting the stabilization from nonlinear frequency shifts.

Recently the stabilization of nonlinear explosive instabilities by nonlinear frequency shifts has been studied and the solution extended beyond the time of explosion [ 1,2]. In situations of such rapid amplitude and phase development we may expect also effects of second dispersive order to be important. The nonlinear interaction of the waves generally introduces frequency shifts of the individual waves and these enter the dispersion function D(o, k) of each wave and thereby influence the equations of motion of the coupled system. This may formally be expressed in terms of the Taylor expansion [3] :

aDj -- a

i

aw at - 22

a2Dj

a2 2

+. . .

Ej =

NL.

where NL stands for nonlinear driving terms, Dj(w, k) is the dispersion function and E. is the electric field. The dispersion function D(w, f ) has the properties that the averaged stored energy of the wave in the linear case is given by Wi = f aDj/awjEjEl~_and that the dispersion relation can be written D(aj, kj) = 0. Con* Work supported by The Norwegian Research Council for Science and Humanities. ** Work supported by The Swedish Natural Science Research Council.

sidering now a nonspecified amplitude Aj = IAj lexp (i#j) we may, by introducing suitable normalizations, study the equations of motion of the real amplitudes Uj and phases I$

auj at

- 6j uj-

j=o,

1,2

a2G. at2

I t 21

a$. au. -I atat I

= u u cos@

kl

(14

(lb)

where the resonance conditions are w0 = w1 t w2 and k. = zl t k2. Furthermore Q = @, - #1 - 42, & are due to third order contributions in the amplitudes and ~j are due to second order dispersive effects. In (lb) the upper signs should be used for mode 0 and the lower signs for modes 1 and 2. These equations can also be derived by a variational principle [4], from the Lagrangian

(2) 21

Volume

48A, number

PHYSICS

1

LETTERS

20 May 1974

stant of motion, ian H given by Hz+

-$

I \

x26 i i g

namely the corresponding

[gj2tu;

$ufu:

Hamilton-

($)‘I

+ uou1u2sin@.

4

Fig. 1. Strongly and Si= -0.01.

oscillative

3

2

1 development

where pi = -0.05

where /3; can be taken to be symmetric without loss of generality. From (1) we may derive the following constants of motion UT (I-2si~)-u:

(l-*~~~)=yx.

(3)

The relations (3) correspond to the Manley-Rowe relations. From the fact that the Lagrangian does not depend explicitly on time, one obtains another con-

22

Due to the second order dispersive terms in (4) the stabilizing influence of the nonlinear frequency shifts may be enhanced or counteracted. If it is enhanced we obtain saturation at a lower average amplitude level, but if it is counteracted we may have unstable modes. In the case shown in fig. 1 all ~5~ = -0.01 and all fl$ = -0.05. We note the strong growth of oscillations which tend to dominate the development. The tendency for oscillations seems to decrease when the constants Si become different and increase with the level of $. Moreover the oscillations may be triggered by a sharp maximum or by the initial conditions. If we choose all fli with the same sign and sign $ = rtrsignpi (the upper sign for j = 0 and the lower sign for j = 1,2) we obtain three explosively unstable modes. In the computations the constants of motion (3) and (4) have been checked numerically and ascertained to remain constants with a high degree of accuracy during the evolution of the system. The frequency of the dispersive oscillations can be estimated from (1) as WD = 6jT1 which turns out to be in excellent agreement with the numerical results.

References [ 1 ] V.N. Oraevskii, H. Wilhelmsson, E.Ya. Kogan and V.P. Pavlenko, Physica Scripta 7 (1973) 217; Phys. Rev. Lett. 30 (1973) 49. [2] J. Weiland and H. Wilhelmsson, Physica Scripta 7 (1973) 222. [3] J. Askne, Int. J. Electronics 32 (1972) 573. [4] K.B. Dysthe, J. Plasma Physics 11 (1974) 63. [5] C.T. Dum and E. Ott, Plasma Phys. 13 (1971) 177. [6] V.N. Tsytovich and H. Wilhelmsson, Physica Scripta 7 (1973) 251.