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Relaxation of nonlinearly coupled modes with different linear damping rates
Volume 45A, number 3
PHYSICS LETTERS
24 September 1973
RELAXATION OF NONLINEARLY COUPLED MODES WITH DIFFERENT LINEAR DAMPING RATES H. WILHELMSSON* Institut für Plasm aphysik der Kernforschungsanlage Jülich GmbH, Assoziation Euratom-KFA, Jiilich Germany Received 27 July 1973 The transition from a highly nonlinear situation for a system of three interacting waves to a state where the disspative effects dominate is studied theoretically for the case were mode damping occurs with different rates.
The well-defined phase (or coherent wave) description assuming monochromatic waves and perfect matching of frequencies (w0 =w1 +w2) and wave vectors (k0=k1 +k2) has been applied in theoretical investigations of stable [1] as well as nonlinearly unstable systems [2—6].The problem of linear dissipation for mutually different damping rates has been studied in the same description for the nonlinearly unstable case [7] also when the damping rates depend on time [8]. The corresponding nonlinear problem with different linear damping rates in the stable case, which should be of considerable interest to plasma physics as well as to modern optics, has however not been discused in literature as far as is known to the author. We may write the set of basic nonlinear equations for such a stable case in the following form:
au1/at +
~0(t)u0 v1(t)u1
au2/at
v2(t)u2
+
+
~1~2
~, —=
= =
~0~2 —
\ U0
uiu2 cos ‘I u0u2 cos ‘1
(la) (lb)
u0u1 cos uØu1\ sin 1,
(ic)
— —,
U1
(2)
U2 /
where the u1 > 0 are normalized moduli of the wave amplitudes, v1(t) linear damping rates and 0~P1~°2~
(3)
with the phase of the jth amplitude. The solutions to eqs. (1,2) are in general oscillatory and bounded. ~.
When the modes of a nonlinear system of waves have mutually different linear damping rates it depends on the degree of nonlinearity if the damping may be characterized as collective or individual. We are interested here in investigating the relaxation from a highly nonlinear situation,where the effective damping of each mode depends on all linear damping coefficients v~,to the linear domain, where the nonlinear terms become gradually less important and the modes are damped essentially by their individual damping coefficients. We use the results from a recent investigation on the evolution of phase uncertainties in nonlinear systems [9] to construct the time for the transition between collective (all modes damped equally) and individual (each mode damped by its own characteristic damping rate v1) damping. We make use of the fact that each value of the damping rate corresponds to a certain spectral width. The solution to the system of equations (1,2) can be constructed in the following form r ~ 1 u~Uj(r)exp (4)
L_f~(t’)dt’i 0
where (J1(T) fulfills the equations where damping terms are absent namely au0/ar = U1 U2 cos (Sa) —
au1/ar=u0u2cos~
(5b)
~u
(Sc)
r=
0
-
*
Permanent address: Institute for Electromagnetic Field
uu
~-
—
(Ui U2 ~(T
cos 1
U0 U2
U0 U1\
J— J)
sin I~
(6)
Theory, Chalmers University of Technology, Göteborg, Sweden.
183
Volume 45A, number 3
PHYSICS LETTERS
for which exact solutions exist [II in the form of elliptic integrals. We have introduced the time-transformation -~
(7)
fexp [~_fot”dt”jdt’.
r
o
o
where =
p~(t)u0+ v1(t)u1
+
v2(t)u2
(8)
24 September 1973
2 is the time average during the process. where uJ We notice that beside damping as expressed by the function (9), which exhibits the transition between collective and individual damping, there is a stretching of the time scale as expressed by relations (7) and (8), which also demonstrate the influence of the seperate p 1. The results obtained here agree with those of the limiting case were all are equal [1].
UO+ul +u2 v0 (t)
U0 + V1 0’) U1
References
+ 1)2 (t) U2
U0+U1+U2 [l[J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127 (1962) 1918. [2] H. Wilhelmsson, J. Plasma Phys. 3 (1969) 215.
Furthermore, the damping is characterized by the
function
h.(t) =
—[P1(t)-—o(t)]
v1(t)
d
(exp {.—t/t1} t),
~
where the time t1. is defined by 2 u1(0).f dt/u~= 1.
fT
(9)
(10)
(1969) 206. [4] B. Coppi, M.N. Rosenbluth and R.N. Sudan, Ann. Phys. (N.Y.) 55 (1969) 207. 15] H. Wilhelmsson, L. Stenflo and F. Engelman. J. Math. Phys. 11(1970)1738. [6] V.N. Tsytovich, Nonlinear effects in plasma (Plenum Press, New York 1970). [71 H. Wilhelmsson, Phys. Scripta (Sweden) 2(1970)113. [8] H. Wilhelmsson, Phys. Rev. A6 (1972) 1973.
(11)
[91 H. Wilhelmsson, Phys. Scripta (Sweden), to be published.
i=0
Hence we may estimate 2 t.
184
1
fT u~(0)}
{~~2
2 ~p
~
2
1
U
1(o)~
.
13] F. Engelmann and H. Withelmsson, Z. Naturforsch 24a
PHYSICS LETTERS
24 September 1973
RELAXATION OF NONLINEARLY COUPLED MODES WITH DIFFERENT LINEAR DAMPING RATES H. WILHELMSSON* Institut für Plasm aphysik der Kernforschungsanlage Jülich GmbH, Assoziation Euratom-KFA, Jiilich Germany Received 27 July 1973 The transition from a highly nonlinear situation for a system of three interacting waves to a state where the disspative effects dominate is studied theoretically for the case were mode damping occurs with different rates.
The well-defined phase (or coherent wave) description assuming monochromatic waves and perfect matching of frequencies (w0 =w1 +w2) and wave vectors (k0=k1 +k2) has been applied in theoretical investigations of stable [1] as well as nonlinearly unstable systems [2—6].The problem of linear dissipation for mutually different damping rates has been studied in the same description for the nonlinearly unstable case [7] also when the damping rates depend on time [8]. The corresponding nonlinear problem with different linear damping rates in the stable case, which should be of considerable interest to plasma physics as well as to modern optics, has however not been discused in literature as far as is known to the author. We may write the set of basic nonlinear equations for such a stable case in the following form:
au1/at +
~0(t)u0 v1(t)u1
au2/at
v2(t)u2
+
+
~1~2
~, —=
= =
~0~2 —
\ U0
uiu2 cos ‘I u0u2 cos ‘1
(la) (lb)
u0u1 cos uØu1\ sin 1,
(ic)
— —,
U1
(2)
U2 /
where the u1 > 0 are normalized moduli of the wave amplitudes, v1(t) linear damping rates and 0~P1~°2~
(3)
with the phase of the jth amplitude. The solutions to eqs. (1,2) are in general oscillatory and bounded. ~.
When the modes of a nonlinear system of waves have mutually different linear damping rates it depends on the degree of nonlinearity if the damping may be characterized as collective or individual. We are interested here in investigating the relaxation from a highly nonlinear situation,where the effective damping of each mode depends on all linear damping coefficients v~,to the linear domain, where the nonlinear terms become gradually less important and the modes are damped essentially by their individual damping coefficients. We use the results from a recent investigation on the evolution of phase uncertainties in nonlinear systems [9] to construct the time for the transition between collective (all modes damped equally) and individual (each mode damped by its own characteristic damping rate v1) damping. We make use of the fact that each value of the damping rate corresponds to a certain spectral width. The solution to the system of equations (1,2) can be constructed in the following form r ~ 1 u~Uj(r)exp (4)
L_f~(t’)dt’i 0
where (J1(T) fulfills the equations where damping terms are absent namely au0/ar = U1 U2 cos (Sa) —
au1/ar=u0u2cos~
(5b)
~u
(Sc)
r=
0
-
*
Permanent address: Institute for Electromagnetic Field
uu
~-
—
(Ui U2 ~(T
cos 1
U0 U2
U0 U1\
J— J)
sin I~
(6)
Theory, Chalmers University of Technology, Göteborg, Sweden.
183
Volume 45A, number 3
PHYSICS LETTERS
for which exact solutions exist [II in the form of elliptic integrals. We have introduced the time-transformation -~
(7)
fexp [~_fot”dt”jdt’.
r
o
o
where =
p~(t)u0+ v1(t)u1
+
v2(t)u2
(8)
24 September 1973
2 is the time average during the process. where uJ We notice that beside damping as expressed by the function (9), which exhibits the transition between collective and individual damping, there is a stretching of the time scale as expressed by relations (7) and (8), which also demonstrate the influence of the seperate p 1. The results obtained here agree with those of the limiting case were all are equal [1].
UO+ul +u2 v0 (t)
U0 + V1 0’) U1
References
+ 1)2 (t) U2
U0+U1+U2 [l[J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127 (1962) 1918. [2] H. Wilhelmsson, J. Plasma Phys. 3 (1969) 215.
Furthermore, the damping is characterized by the
function
h.(t) =
—[P1(t)-—o(t)]
v1(t)
d
(exp {.—t/t1} t),
~
where the time t1. is defined by 2 u1(0).f dt/u~= 1.
fT
(9)
(10)
(1969) 206. [4] B. Coppi, M.N. Rosenbluth and R.N. Sudan, Ann. Phys. (N.Y.) 55 (1969) 207. 15] H. Wilhelmsson, L. Stenflo and F. Engelman. J. Math. Phys. 11(1970)1738. [6] V.N. Tsytovich, Nonlinear effects in plasma (Plenum Press, New York 1970). [71 H. Wilhelmsson, Phys. Scripta (Sweden) 2(1970)113. [8] H. Wilhelmsson, Phys. Rev. A6 (1972) 1973.
(11)
[91 H. Wilhelmsson, Phys. Scripta (Sweden), to be published.
i=0
Hence we may estimate 2 t.
184
1
fT u~(0)}
{~~2
2 ~p
~
2
1
U
1(o)~
.
13] F. Engelmann and H. Withelmsson, Z. Naturforsch 24a