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The scaling reduction of the three-wave resonant system and the Painlevé VI equation
Volume
115, number
PHYSICS
7
LETTERS
THE SCALING REDUCTION OF THE THREE-WAVE AND THE PAINLEVi VI EQUATION A.S. FOKAS Dipnrtimento and INFN,
Received
‘, R.A. LEO, L. MARTINA
di Fish
dell’Uniuersith
and
28 April 1986
A
RESONANT
SYSTEM
G. SOLIANI
73100 Lmce, Italy
Gruppo di Lecce, Serione di Bari, Bari. Italy
7 November
1985; accepted
in revised form 17 February
1986
The scaling invariant solutions of the three-wave resonant system in one spatial and one temporal dimension satisfy a system of three first-order nonlinear ordinary differential equations. These equations can be reduced to one second-order equation quadratic in the second derivative. This equation is outside the class of equations classified by Painleve and his school. However. it is a special case of an equation recently found to be related via a one-to-one transformation to the Painleve VI equation
1. Introduction. Given a partial differential equation (PDE) there exists an algorithmic approach of finding all its Lie-point symmetries [ 1,2], i.e. of finding all infinitesimal generators (elements of a Lie algebra) of those groups of transformations which leave the given equation invariant. Such groups are called admissible. A number of Lie-point symmetries can be found by inspection. For example, the Korteweg-de Vries (KdV) equation z+ + uxxx + 6uu, = 0 is invariant under x-translation (x’ = x t aI), t-translation (t’ = t t a2), and appropriate scaling (x’ = (1 t 03)x, t’ = (1 + 03)3 t, u’ = (1 + (Ye)- 2u). These groups are uniquely characterized by the Lie operators X, = d/ax, X2 = a/at, X3 = xa/ax t 3talat - 2ualau. An admissible group (under certain conditions [ 11): (i) Maps solutions of the given PDE among themselves. (ii) Characterizes solutions which are mapped to themselves under the action of the group. These invariant or similarity solutions also satisfy an equation with one less independent variable than the original PDE. For example, the scaling invariant solutions of the Kortewegde Vries equation also solve U”’ + 6UU’ - (2U t zu’) = 0, U = (3t)-2&J(z), z = x(3t)-113. ’ Permanent address: Department of Mathematics and Computer Science and Institute for Nonlinear Studies, Clarkson University, Potsdam, NY 13676, USA.
0.3759601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Invariant solutions are of interest because: (i) If the boundary conditions are also invariant under the ad. missible group then an invariant solution is the unique solution of the given PDE. (ii) Generically, asymptotic states of a PDE usually satisfy invariant solutions. (This interesting phenomenon, although observed in a vast number of systems, has not been, in the author’s opinion, adequately explained.) For example, a large number of equations, including the KdV, possess the property that as t + 00,generic initial data decompose into a number of solitary waves; solitary waves are the invariant solutions under X1 t fix2 [3]. Similarly, scaling invariant solutions are important in the large-t behavior of KdV, modified KdV, sine-Gordon [4]. The above discussion refers to any PDE, not necessarily exactly solvable. However, if an equation is integrable, Ablowitz, Ramani and Segur [S] have conjectured that its invariant solutions (within a change of variables) posses the Painleve property, i.e. their only movable singularities are poles. Hence they satisfy equations of Painleve type, and in particular if they are of the form w” = F(w’, w, z), where F is rational in w’, algebraic in w and locally analytic in z, they must be among the equations classified by Painled [6] and his school [7] at the turn of the century. We remind the reader that, within a Mobius transformation, there exist fifty such equations. Distinguished 329
Volume
115, number
7
PHYSICS
among them are Painleve I-Painleve VI (P I-P VI), since any other of the fifty equations can either be integrated in terms of known functions or can be reduced to one of these six equations. From the above discussion it follows that: (i) Invariant solutions of physically important PDEs are in general physically interesting, at least for the characterization of the asymptotic states of the underlying physical system. (ii) Painleve equations should be of physical interest since the invariant solutions of integrable PDEs are in general of Painleve type. Indeed, P II and special cases of P III, P IV may be obtained from the similarity reduction of the modified KdV, of the sine-Gordon and of the nonlinear Schrodinger equations, respectively. Furthermore, Painled equations appear in a number of other physical problems, see e.g. refs. [8,11]. Painled equations are not only of historical and physical significance. They are also mathematically interesting since they can be investigated by exact methods: (i) For certain choices of their parameters, P IIP VI admit particular exact solutions which are either rational functions or are related (via repeated differentiations and multiplications) to Airy [6,12], Bessel [ 13 1, Weber-Hermite [ 141, Whittaker [ 151, hypergeometric [ 161 functions, respectively. (ii) The general solutions of P II-P VI can be obtained via solving appropriate Riemann-Hilbert problems (for P II see refs. [ 17,181, for P III-P V see ref. [19], for P VI see ref. [20]). Series representations of these solutions are given in ref. [2 11. A fundamental role in the exact treatment of the Painleve equations is played by certain transformations which map solutions of a given Painleve to solutions of the same PairtIed but with different choice of the parameters. Such transformations for P II-P VI were given in refs. [12-161 respectively. Finding such a transformation for P VI necessitated the introduction of an auxiliary equation (see (2) below). This equation is quadratic in the second derivative and is thus outside the original classification of Painled. However, it was shown in ref. [16] that it can be mapped to P VI. This paper is concerned with the scaling invariant solutions of the three wave resonant interaction equations in one spatial and one temporal dimension. We expect that such solutions are important in the 330
LETTERS
A
28 April 1986
large t asymptotic limit. We find that the scaling invariant solutions satisfy a certain second order ordinary differential equation which is precisely (within a choice of parameters) the auxiliary equation mentioned above. Thus using the results of refs. [ 16,201 the scaling invariant solutions of the three-wave resonant interactions can be obtained via solving a RiemannHilbert problem. 2. The three-wve resonant interactions and Painlevt VI. For the reader’s convenience we first recall theorem 1 of ref. [16] (see also theorem 6.1 in ref. [22] ). l%eorem (Fohzs and Yortsos). Let u(z; a, j3,y, 6) be a solution of P VI,
$=l
(i+L+-
2 v
1 12 v-z ) ”
u-l
,u(u-l)(u~z)
and let @(z; K~, 1)2$
(
+_!-Or
u-z
1
o+pz+y(z-1)+6z(z-l) (
z2(z - 1)2
(z -
L+-- ’ z-l Z
_
h,
=;
p,
u2 v)
(u - 1)2
(u - z)2 1 ’ (1)
be a solution of the equation
(rp’2 + ;(;:;;2)y2
,
(2)
where
n1~,,t(3z-1)o’t2~zt;~-~~2 (3a) 2z(z - 1)
z(z - I)2
’
IaB2+.l@ tv,
(3b)
\k~(ztl)@t+;y(ztl)t~~(Z-1),
(3c)
and the parameters K , tions K=&
--Cl1
-
H=(4/K)(: “1
=
1,
A, p,
h=q
are defined by the equa-
+k$
>
V = 26 - 1 + ($A + ;K)2
-T-f%
(2a)“2,
v
pr = (-2/3)0)“2 .
,
(4)
Assume that @#O,
Kfo,
V#o.
(5)
Then there exists the following one-to-one correspondence between solutions of eqs, (1) and (2):
Q=z<+~-‘+l u
U+(htKtl)Z
2(z - 1)
--1 u
2(2-l)
{A-
z+l z-l
4; +$o, u=
-[(Z
t
(6)
1)/Z]@‘+ -2@“Z
[2K/Z(Z tI/Z2
28 April 1986
PHYSICS LETTERS A
Volume 115, number 7
-
I)]@-(Z
t K(Z t 1)@/Z2(Z
-
1)2i2/\k
Proposition. Eq. (13) can be reduced to eq. (2) in the special case that (2) has only one free parameter v = -pw2 and k, X, M are given by k2 = 4v ,
A=/.l=O.
(14)
The above reduction
involves
’
1)
(7)
+~(b)+WY)+tiZ)Y
(15)
C+_Y-+a,
where For the sake of definiteness we consider the threewave resonant interactions in the case of explosive instability [23,24] : (&!j>t + Cj
j,k,l=
jZk#l,
1,2,3,
(8)
\k,
= (0 _
l)1/2m:/2$l
,
$
=
-.Z..L
(w2)1/2 ’ C3 - Cl rJi----, C2 -cl
w2 G [(c3 - cl)(c3
- c2)mlm2m3]-1
(16)
where ui(x, t) are the complex amplitudes of the wave packets, Cj are their group velocities and * denotes complex conjugate. Without loss of generality we assume Cl cc,
(9)
The initial value problem associated with eqs. (8), for initial data decaying to zero sufficiently fast as Ix] + m has been considered in refs. [23,25], using the inverse scattering transform. Eqs. (8) are invariant under x-translation, f-translation, and appropriate scaling. Hence the following is an admissible similarity reduction: d
cj-c
d{ Uj - bjS
mj(UE - bii) (V; - b$) ’ jfkfl,
1,2,3,
(10)
’
c3tc1-2c2
p&
c3 -cl (17)
c3 t Cl - 2c2 )
and p is a constant of integration. To derive the above note that eqs. (13) imply ml(c2 - S)(c3 - {)+,GQ~ =m2(c3 =
-
S)(cl
-
05245~
L
m3(cl - {)(C2
-
{)+3*3f
= \k,+2*3
.
Hence (u - l)ml\irl
_-=
j,k,l=
(C3 - C2)(C3 - Cl)
a&
- orn,Gi t rn3+g + p = 0.
Solve the above for $3 and substitu!e in (13a), (13b). Then change variables from +_I({), *2(t), to \kl(~), \k2~), where q2 = u1/2m1/2*2, to obtain
where (Gxft,
Uk & a&$
t b,x/t ,
\k,,
= [w\k2/rO
- l)] (\k; - \k: - PY2
mj H aj/a%T, (11) 92, = [2w\k&V2-
and ak, bk are arbitrary constants. In the following we consider only imaginary solutions and assume that mj are real. Hence, letting +k 2 i(ek - c)/(uk we obtain ^ ^ ^ (*& = *2*3/ml(c2
- b&
,
- 5)@3 - 5))
(13b), (13~) follow by cyclic permutation.
l)] <\k; - \k; - P)“2.
,
(184 08b)
Eqs. (18) imply that (19)
(12)
(W
t (1/2w) [(!Pf t p)2w2 + 4_Y2(Y- 1)2u:,] Differentiating the above with respect toy (19) it follows that
1’2 . and using
331
,
Volume
115, number
w$+l
7
PHYSICS
[(!IJf t p)2w2 + 4&Y
= w+I?; + *1,
+ p)*l
Iv2ti
-
+ 2\k,,,y”Q
021, .
- l)%$
l/2
- 1)2
(20)
This research was supported in part by the National Science Foundation under grant number MCS-8202 117, the Office of Naval Research under grant number N00014-76C09867. References
[2] [3]
[4] [S]
332
A
Group properties of differential equations (Translated from the Russian version by G.W. Blurnan), original publication in 1962. G.W. Bluman and J.D. Cole, Similarity methods for differential equations (Springer, Berlin, 1974). J. Bona, J. Albert and D. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, preprint (1985). M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (SIAM, Philadelphia, 1981). M.J. Ablowitz, A. Ramani and H. Segur, Lett. Nuovo Cimento 23 (1978) 333; J. Math. Phys. 21 (1980) 715; 1006.
28 April 1986
161 P. Painleve, Bull. Sot. Math. France 28 (1900) 214; Acta Math. 25 (1902) 1. Acta. Math. 33 (1909) 1. B.M. McCoy and T.T. Wu, Phys. Rev. Lett. 31 (1973) 1409; T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Phys. Rev. B 13 (1976) 316. 191 M. Jimbo, T. Miwa, Y. Mori and M. Sato, Physica D I (1980) 80; M. Jimbo and T. Miwa, Proc. Japan Acad. A 56 (1980) 405. IlO1 J.W. Miles, Proc. R. Sot. A 361 (1978) 277. reduction of the Ernst’s 1111 C. Cosgrove, An appropriate equation yields P VI, private communication. [I21 N.P. Erugin, Dokl. Akad. Nauk. BSSR 2 (1958); N.A. Lukashevich, Ditf. Urav. 7 (1971) 1124. Diff. Urav. 3 (1967) 994; 1131 N.A. Lukashevich, V.I. Gromak, Diff. Urav. 11 (1975) 373. Diff. Urav. 3 (1967) 395;Abstract 1141 N.A. Lukashevich, of Doctoral Thesis, Kiev (1971). Diff. Urav. 4 (1968) 732; [151 N.A. Lukashevich, V.I. Gromak, Diff. Urav. 12 (1976) 740. [I61 A.S. Fokas and Y.C. Yortsos, Lett. Nuovo Cimento 30 (1981) 539. [I71 H. Flaschka and AC. Newell, Commun. Math. Phys. 76 (1980) 65. 1181 AS. Fokas and M.J. Ablowitz, Commun. Math. Phys. 91 (1983) 381. [I91 A.S. Fokas, U. Mugan and M.J. Ablowitz, On the initial value problem of P II-P V (preprint), Clarkson University, Potsdam, NY 13676. [ZO] C. Cosgrove, The monodromy problem and connection formulas for Painleve’s sixth equation (1986), Clarkson University, Potsdam, NY 13676. [21] M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto Univ. 15 (1979) 201; Proc. Japan. Acad. A 55 (1979) 267; In: Lecture notes in physics, Vol. 126 (Springer, Berlin, 1980); M. Jimbo, T. Miwa and K. Ueno, Physica D 2 (1981) 306; M. Jimbo and T. Miwa, Physica D 2 (1981) 407; 4 (1981) 47. [22] A.S. Fokas and M.J. Ablowitz, J. Math. Phys. 23 (1982) 2033. [23] V.E. Zakharov and S.V. Manakov, Sov. Phys. JETP 42 (1976) 842. [24] D. Kaup, A.H. Reiman and A. Bers, Rev. Mod. Phys. 51 (1979) 275. [25] D. Kaup, Phys. D 1 (1980) 45.
[71 B. Gambier, (81 E. Barouch,
Eq. (20) under the transformationy + z, Jr, -+ $I reduces to a special case of eq. (2). Remarks. (1) Since we are interested only in real solutions v > 0 or p < 0. (2) Using eqs. (4) it follows that the similarity reduction of the three wave resonant interactions can be mapped to two different P VI equations, with parameters {(Y,/3, y, 6 } given by { f(v - $)2, -i(v - $)2, O,a}and {~(~+f)~,--~(v+$)~,O,~}. (3) The formalism presented here also applies to the non-explosive case (i.e. equations like (8) but with a sign change in some interaction coefficients). However, we expect that in the explosive, as opposed to non-explosive case, the solutions of eqs. (13) might possess a pole singularity for finite values of the similarity variable. (4) The treatment of complex as opposed to purely imaginary solutions remains open.
[l] L.V. Ovsjannikov,
LETTERS
115, number
PHYSICS
7
LETTERS
THE SCALING REDUCTION OF THE THREE-WAVE AND THE PAINLEVi VI EQUATION A.S. FOKAS Dipnrtimento and INFN,
Received
‘, R.A. LEO, L. MARTINA
di Fish
dell’Uniuersith
and
28 April 1986
A
RESONANT
SYSTEM
G. SOLIANI
73100 Lmce, Italy
Gruppo di Lecce, Serione di Bari, Bari. Italy
7 November
1985; accepted
in revised form 17 February
1986
The scaling invariant solutions of the three-wave resonant system in one spatial and one temporal dimension satisfy a system of three first-order nonlinear ordinary differential equations. These equations can be reduced to one second-order equation quadratic in the second derivative. This equation is outside the class of equations classified by Painleve and his school. However. it is a special case of an equation recently found to be related via a one-to-one transformation to the Painleve VI equation
1. Introduction. Given a partial differential equation (PDE) there exists an algorithmic approach of finding all its Lie-point symmetries [ 1,2], i.e. of finding all infinitesimal generators (elements of a Lie algebra) of those groups of transformations which leave the given equation invariant. Such groups are called admissible. A number of Lie-point symmetries can be found by inspection. For example, the Korteweg-de Vries (KdV) equation z+ + uxxx + 6uu, = 0 is invariant under x-translation (x’ = x t aI), t-translation (t’ = t t a2), and appropriate scaling (x’ = (1 t 03)x, t’ = (1 + 03)3 t, u’ = (1 + (Ye)- 2u). These groups are uniquely characterized by the Lie operators X, = d/ax, X2 = a/at, X3 = xa/ax t 3talat - 2ualau. An admissible group (under certain conditions [ 11): (i) Maps solutions of the given PDE among themselves. (ii) Characterizes solutions which are mapped to themselves under the action of the group. These invariant or similarity solutions also satisfy an equation with one less independent variable than the original PDE. For example, the scaling invariant solutions of the Kortewegde Vries equation also solve U”’ + 6UU’ - (2U t zu’) = 0, U = (3t)-2&J(z), z = x(3t)-113. ’ Permanent address: Department of Mathematics and Computer Science and Institute for Nonlinear Studies, Clarkson University, Potsdam, NY 13676, USA.
0.3759601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Invariant solutions are of interest because: (i) If the boundary conditions are also invariant under the ad. missible group then an invariant solution is the unique solution of the given PDE. (ii) Generically, asymptotic states of a PDE usually satisfy invariant solutions. (This interesting phenomenon, although observed in a vast number of systems, has not been, in the author’s opinion, adequately explained.) For example, a large number of equations, including the KdV, possess the property that as t + 00,generic initial data decompose into a number of solitary waves; solitary waves are the invariant solutions under X1 t fix2 [3]. Similarly, scaling invariant solutions are important in the large-t behavior of KdV, modified KdV, sine-Gordon [4]. The above discussion refers to any PDE, not necessarily exactly solvable. However, if an equation is integrable, Ablowitz, Ramani and Segur [S] have conjectured that its invariant solutions (within a change of variables) posses the Painleve property, i.e. their only movable singularities are poles. Hence they satisfy equations of Painleve type, and in particular if they are of the form w” = F(w’, w, z), where F is rational in w’, algebraic in w and locally analytic in z, they must be among the equations classified by Painled [6] and his school [7] at the turn of the century. We remind the reader that, within a Mobius transformation, there exist fifty such equations. Distinguished 329
Volume
115, number
7
PHYSICS
among them are Painleve I-Painleve VI (P I-P VI), since any other of the fifty equations can either be integrated in terms of known functions or can be reduced to one of these six equations. From the above discussion it follows that: (i) Invariant solutions of physically important PDEs are in general physically interesting, at least for the characterization of the asymptotic states of the underlying physical system. (ii) Painleve equations should be of physical interest since the invariant solutions of integrable PDEs are in general of Painleve type. Indeed, P II and special cases of P III, P IV may be obtained from the similarity reduction of the modified KdV, of the sine-Gordon and of the nonlinear Schrodinger equations, respectively. Furthermore, Painled equations appear in a number of other physical problems, see e.g. refs. [8,11]. Painled equations are not only of historical and physical significance. They are also mathematically interesting since they can be investigated by exact methods: (i) For certain choices of their parameters, P IIP VI admit particular exact solutions which are either rational functions or are related (via repeated differentiations and multiplications) to Airy [6,12], Bessel [ 13 1, Weber-Hermite [ 141, Whittaker [ 151, hypergeometric [ 161 functions, respectively. (ii) The general solutions of P II-P VI can be obtained via solving appropriate Riemann-Hilbert problems (for P II see refs. [ 17,181, for P III-P V see ref. [19], for P VI see ref. [20]). Series representations of these solutions are given in ref. [2 11. A fundamental role in the exact treatment of the Painleve equations is played by certain transformations which map solutions of a given Painleve to solutions of the same PairtIed but with different choice of the parameters. Such transformations for P II-P VI were given in refs. [12-161 respectively. Finding such a transformation for P VI necessitated the introduction of an auxiliary equation (see (2) below). This equation is quadratic in the second derivative and is thus outside the original classification of Painled. However, it was shown in ref. [16] that it can be mapped to P VI. This paper is concerned with the scaling invariant solutions of the three wave resonant interaction equations in one spatial and one temporal dimension. We expect that such solutions are important in the 330
LETTERS
A
28 April 1986
large t asymptotic limit. We find that the scaling invariant solutions satisfy a certain second order ordinary differential equation which is precisely (within a choice of parameters) the auxiliary equation mentioned above. Thus using the results of refs. [ 16,201 the scaling invariant solutions of the three-wave resonant interactions can be obtained via solving a RiemannHilbert problem. 2. The three-wve resonant interactions and Painlevt VI. For the reader’s convenience we first recall theorem 1 of ref. [16] (see also theorem 6.1 in ref. [22] ). l%eorem (Fohzs and Yortsos). Let u(z; a, j3,y, 6) be a solution of P VI,
$=l
(i+L+-
2 v
1 12 v-z ) ”
u-l
,u(u-l)(u~z)
and let @(z; K~, 1)2$
(
+_!-Or
u-z
1
o+pz+y(z-1)+6z(z-l) (
z2(z - 1)2
(z -
L+-- ’ z-l Z
_
h,
=;
p,
u2 v)
(u - 1)2
(u - z)2 1 ’ (1)
be a solution of the equation
(rp’2 + ;(;:;;2)y2
,
(2)
where
n1~,,t(3z-1)o’t2~zt;~-~~2 (3a) 2z(z - 1)
z(z - I)2
’
IaB2+.l@ tv,
(3b)
\k~(ztl)@t+;y(ztl)t~~(Z-1),
(3c)
and the parameters K , tions K=&
--Cl1
-
H=(4/K)(: “1
=
1,
A, p,
h=q
are defined by the equa-
+k$
>
V = 26 - 1 + ($A + ;K)2
-T-f%
(2a)“2,
v
pr = (-2/3)0)“2 .
,
(4)
Assume that @#O,
Kfo,
V#o.
(5)
Then there exists the following one-to-one correspondence between solutions of eqs, (1) and (2):
Q=z<+~-‘+l u
U+(htKtl)Z
2(z - 1)
--1 u
2(2-l)
{A-
z+l z-l
4; +$o, u=
-[(Z
t
(6)
1)/Z]@‘+ -2@“Z
[2K/Z(Z tI/Z2
28 April 1986
PHYSICS LETTERS A
Volume 115, number 7
-
I)]@-(Z
t K(Z t 1)@/Z2(Z
-
1)2i2/\k
Proposition. Eq. (13) can be reduced to eq. (2) in the special case that (2) has only one free parameter v = -pw2 and k, X, M are given by k2 = 4v ,
A=/.l=O.
(14)
The above reduction
involves
’
1)
(7)
+~(b)+WY)+tiZ)Y
(15)
C+_Y-+a,
where For the sake of definiteness we consider the threewave resonant interactions in the case of explosive instability [23,24] : (&!j>t + Cj
j,k,l=
jZk#l,
1,2,3,
(8)
\k,
= (0 _
l)1/2m:/2$l
,
$
=
-.Z..L
(w2)1/2 ’ C3 - Cl rJi----, C2 -cl
w2 G [(c3 - cl)(c3
- c2)mlm2m3]-1
(16)
where ui(x, t) are the complex amplitudes of the wave packets, Cj are their group velocities and * denotes complex conjugate. Without loss of generality we assume Cl cc,
(9)
The initial value problem associated with eqs. (8), for initial data decaying to zero sufficiently fast as Ix] + m has been considered in refs. [23,25], using the inverse scattering transform. Eqs. (8) are invariant under x-translation, f-translation, and appropriate scaling. Hence the following is an admissible similarity reduction: d
cj-c
d{ Uj - bjS
mj(UE - bii) (V; - b$) ’ jfkfl,
1,2,3,
(10)
’
c3tc1-2c2
p&
c3 -cl (17)
c3 t Cl - 2c2 )
and p is a constant of integration. To derive the above note that eqs. (13) imply ml(c2 - S)(c3 - {)+,GQ~ =m2(c3 =
-
S)(cl
-
05245~
L
m3(cl - {)(C2
-
{)+3*3f
= \k,+2*3
.
Hence (u - l)ml\irl
_-=
j,k,l=
(C3 - C2)(C3 - Cl)
a&
- orn,Gi t rn3+g + p = 0.
Solve the above for $3 and substitu!e in (13a), (13b). Then change variables from +_I({), *2(t), to \kl(~), \k2~), where q2 = u1/2m1/2*2, to obtain
where (Gxft,
Uk & a&$
t b,x/t ,
\k,,
= [w\k2/rO
- l)] (\k; - \k: - PY2
mj H aj/a%T, (11) 92, = [2w\k&V2-
and ak, bk are arbitrary constants. In the following we consider only imaginary solutions and assume that mj are real. Hence, letting +k 2 i(ek - c)/(uk we obtain ^ ^ ^ (*& = *2*3/ml(c2
- b&
,
- 5)@3 - 5))
(13b), (13~) follow by cyclic permutation.
l)] <\k; - \k; - P)“2.
,
(184 08b)
Eqs. (18) imply that (19)
(12)
(W
t (1/2w) [(!Pf t p)2w2 + 4_Y2(Y- 1)2u:,] Differentiating the above with respect toy (19) it follows that
1’2 . and using
331
,
Volume
115, number
w$+l
7
PHYSICS
[(!IJf t p)2w2 + 4&Y
= w+I?; + *1,
+ p)*l
Iv2ti
-
+ 2\k,,,y”Q
021, .
- l)%$
l/2
- 1)2
(20)
This research was supported in part by the National Science Foundation under grant number MCS-8202 117, the Office of Naval Research under grant number N00014-76C09867. References
[2] [3]
[4] [S]
332
A
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Eq. (20) under the transformationy + z, Jr, -+ $I reduces to a special case of eq. (2). Remarks. (1) Since we are interested only in real solutions v > 0 or p < 0. (2) Using eqs. (4) it follows that the similarity reduction of the three wave resonant interactions can be mapped to two different P VI equations, with parameters {(Y,/3, y, 6 } given by { f(v - $)2, -i(v - $)2, O,a}and {~(~+f)~,--~(v+$)~,O,~}. (3) The formalism presented here also applies to the non-explosive case (i.e. equations like (8) but with a sign change in some interaction coefficients). However, we expect that in the explosive, as opposed to non-explosive case, the solutions of eqs. (13) might possess a pole singularity for finite values of the similarity variable. (4) The treatment of complex as opposed to purely imaginary solutions remains open.
[l] L.V. Ovsjannikov,
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