- Email: [email protected]
On the dynamics of scalar wave collapse
23 February 1998
PHYSICS
ELSEMER
LETTERS
A
Physics Letters A 239 ( 1998) 46-50
On the dynamics of scalar wave coiiapse E.M. Maslov **I,A.G. Shagalov bs a fmtitu~eof Terrestrial ~ag~edsm,
ionosphere and Rad~owave Propagafion of the Russian Academy of Sciences (IZM~~N~, Troifsk, Moscow Region 142092. Russia b institute of Metal Physics of the Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya 18, GSP-170, Ekaterinburg, 620219, Russia
Received 5 November 1997; accepted for publication 2 December 1997 Communicated by V.M. Agranovich
Abstract The collapse in the nonlinear Klein~ordon equation is considered in a general way for a sufficiently wide class of nonline~ties. A ~ffe~ntial in~u~ity making it possible to estimate the collapse dynamics is derived. As examples, the estimates are obtained for some nonline~ties having physical interest. @ 1998 Published by Elsevier Science B.V. Keywords: Multidimensional solitons; Collapse
1. Introduction The explosive increase of a field in a finite space region is a widespread manifestation of instability in nonlinear dynamical systems. In the physics literature this phenomenon is usually referred to as wave collapse, since in most cases the increase of a field amplitude is accompanied by a contraction of a field distribution. The corresponding dynamical structures (frequently called collapsons, explosons, etc.) are essentially nonlinear and, as a rule, become singular in a finite time. The best-investigated dynamical systems in which the collapse phenomenon occurs are the Zakharov equatians and the nonlinear Schrtidinger equation with a power nonlinearity. In these cases a number of both numerical and self-similar solutions describing a field near singul~ity have been found (see, e.g., ’ Corresponding author. E-mail: [email protected] 2 E-mail: arkadi.shagaiov9usu.m
Refs. [ l-41 ), as well as the sufficient conditions for the initial field distributions under which the collapse occurs have been derived [ $6 ] _ In the relativistic case, in particular, for rhe nonlinear Klein-Gordon equation, the collapse is less well understood. In Refs. [7-91 we have obtained the sufficient conditions for the collapse in the nonlinear Klein-Gordon equation with a logarithmic nonlinearity and have also investigated numerically the collapse dynamics. The corresponding sufficient conditions in the c$~model have been found in Ref. [ IO]. In the present Letter we make an attempt to investigate in a general way the collapse in the real nonlinear Klein-Gordon equation, &t - W -t lJ’ (4) = 0,
(1)
for a sufficiently wide class of nonline~ities. Namely, we only suppose that U(4) - $4 U’ (#) is a convex non-negative differentiable function of tb2. It turns out that for such potentials the approach developed in
037%9601/98/$19.00 @ 1998 Published by Elsevier Science B.V. All rights reserved. PfI SO375-9601(97)00958-4
E.M. Maslov, A.G. Shagalov/Physics
Refs. [7-91 can be generalized giving a useful tool for investigation of dynamics of the collapse. As an illustration, we obtain some estimates for collapse dynamics in the model with a polynomial nonlinearity and in the sinh-Gordon model.
Letters A 239 fl998)
41
46-50
Taking into account the Cauchy-Schwarz inequality A: Q 4AB, from (7) we obtain the nonlinear differential inequality A,t - A;/A
- 4
JF(@‘)
dr i- 4E 3 0
(8)
W” 2. The basic inequality
supplemented with the initial conditions
Let us consider a lump of a real scalar field possessing a finite energy
A(0)
E=
[~(#t)2+~(V~)2+U(~)]dr
s ‘F?’
(the integration is over space IX”;the field 4 (r, the potential U {$) are ficiently fast at infinity). lump as sb(r,O)=f(r),
(2)
the n-dimensional Euclidean t) , its spatial derivatives, and assumed to be vanishing sufDenote the initial state of the
SSt(r,O)=g(r).
(3)
These two functions determine the value of the energy E. On the other hand, by analogy with Refs. [ 7-91, we can recast Eq. (2) into
where F(#2) = U(4) - $$ W$)
(5)
is assumed to be a function of #2. Notice, that F(0) = 0. Considering E as a known quantity and denoting A(t) = /42(r,t)dr. W”
B(t)
=
J&(r, R"
t) dr
J
F(+2)dr=4(B-E).
118”
f”dr,
A,(O) = 2
J
fgdr.
(9)
W”
This inequality is the generalization of the one obtained in Ref. [7]. The function F(4*) is determined by the potential U( 4). Conversely, for a given F( +2) expression (5) can be regarded as a differential equation determining U(4). Let, for example, F( (b2) = ( A/2)#2 (A > 0). Then from Eq. (5) one finds U(#) = -(A/2)+2 log @2 + const x $2. This case has been investigated in detail in Refs. [ 7-9]. In particular, the conditions for initial data (3) have been found there under which the field amplitude squared begins to increase extremely fast (N exp( At2) ) in a finite space domain (N h-‘/2). In the general case one fails to find conditions of this type. The evident exception is the states with E < 0. As is seen from (8)) in this case for any F(#‘) 3 0 any initiat data lead to the unlimited increase of A(t), in agreement with the results of Refs. [ 7-91. In the present Letter we proceed from the assumption that the collapse takes place a priori, and that it is evolving in a finite space domain. To investigate its dynamics we invoke the following simple physical considerations. Define a center and a radius squared of the collapsing lump respectively as
(6)
1 A
JrqS2 dr,
W”
{Ar2>= ;
J (r- al242dr,
( 10)
HP”
we rewrite (4) in the form Al,-4
J
W”
a=-
J
=
(7)
where A is defined in (6). Suppose that (Ar2)‘/2 as a function of time has a least upper bound R, R = sup,(Ar 2) ‘I2 . Physically, this means that at all times the increase of the field amplitude occurs mainly in
48
E.M. Maslov, A.G. Shagalov/Physics
the ~-dimensional ball ll8: of the fixed finite radius R centered at a. Define a function m(t) as m=
-&&ti2dr.
Letters A 239 (1998) 46-N
that for the weII-localized collapsing states with sufficiently large amplitude the values of R are not too large (of the order of half the initial lump), while ,u is close to the unity. 3. The effective potential and comparison
From the definition of B;t, it is clear that for the collapsing states 0 < ,u 6 m(t) < 1, where fi = inf, m. Thus we can write
theorems
With the help of the transformation eq(r) = @2-‘A(t),
r z &?-‘/“t,
(12) I = 4@+-‘E, Let now F( 42) (5) be a non-negative convex differentiable function of 4*. Since F(0) = 0, this implies that F(42) and its derivative with respect to +2 increase monotonically with increasing +*. On the other hand, because F‘(42) is convex, the Jensen inequality is valid (see, e.g., Ref. [ 111). Thus, taking into account (12) we have
< $
qJ
F(42)dr
< ;
IT+*) dr, JW”
(13)
3(eq)
= 4,~&*f”F(,u~~‘A)
(16)
inequality (15) takes the Newtonian form qn > -m/dq
(17)
with the effective potential
U(q) = 4
3( e*) e-* dq.
e-4 J
(18)
Consider the properties of the potential (18) taking into account the above suppositions concerning F(t$2). In accordance with them 3(eq) is a convex non-negative differentiable function of e4 increasing monotonically, with 3(O) = 0. Therefore, from ECq.( lS> we have (omitting an arbitrary constant)
where
n=
J
dr=
.rm/* R”.
l-(42
+ 1)
U(q) --+ -Eew4 - 3’(O) q - %3”(O) e4 - . . . (14)
(4 --) -co)9 U(q)/q
Returning to (8) with this result we obtain A,, - A;/A
c 4E > 4f2F( (p/%)A).
(151
This inequality supplemented with the initial conditions (9) serves as a basis for the subsequent analysis. We call attention to the following. The r.h.s. of ( 15) contains two unknown constants: 0 < p < 1 and 0 < R < 00. Their exact values can be calculated only knowing the solution +(t, t) for all t > 0. This prevents the finding of the sufficient conditions for the coIlapse in the spirit of Refs. [ 7-91. Nevertheless, if the collapse takes place, its dynamics can be investigated on the basis of the inequality (15) supposing only the existence of the above constants. Notice
--, --oo
(4 --+ +cQ)T
(1%
where primes denote the derivatives with respect to x = ee. The corresponding “force” in the r.h.s. of ( 17) has the form -d U/dq = e-9[ 3( eq) -&I . It is important that for & 2 0 this force increases monotonically with increasing q. Indeed, taking the derivative with respect to q one has -d2U/dq2 = e-q[ l+eq3’( eQ) 3( eq) J . This expression is positive, because any convex function 3(x) such that 3(O) = 0 satisfies the inequality x3’(x) > 3 (n) (x > 0). For & > 0 the force is zero at q = qo, where 3(eqo) = 8. At this point the potential has a maximum. For & < 0 the regions of monotonic variation of the force are separated by a point q* de~rmined From the equation e**3’(e@ ) - 3( eq*) = -&. The force decreases
where s > 1 is an integer, A > 0. This potential belongs to the class of polynomial ones. Such potentials occur, for example, in the theory of water wave generation by wind [ lo], and, for negative A, in condensed matter physics [ 131, in particle physics and cosmology [ 141 fd4 theory). With the potential (2 I f we have F(#?)
= h(s - 1)&S,
and hence 3(e9> =4ff - i)A&Pe4”. From ( 18) we thus obtain Fig. I. The qualitative behavior of the effective potential ( 18) for various energies of the collapsing lump.
when --00 < 4 < q, and increases when q* < 4 < co with increasing q. At 4 = qr the corresponding curve U(g) has an inflection point. So, we have generally the picture presented in Fig. 1. In the collapse A - eq tends to -+-co. To obtain estimates for its rate let us consider the motion of a Newtonian particle. say Q. obeying, instead of ( 17), the equation & = -d U( @ ,/d& Suppose the initial conditions (e.g., for T = 0) are so chosen that the 4particle moves in the region of monotonic increase of the force. Then, in accordance with the comparison theorems f 121, the inequality q(7) > &T) is valid for 0 < T < rc provided the initial conditions for the g-particle. and the &particle coincide. Obviously, the collapse time 7-Ccan be estimated as
U(q) = -&e-q
A0
>
e8(7Mo)
(23)
where
(24)
4. Examples As a first example we take U(ss) = -,c$~” + const x &,
(22)
Now we use the considerations given at the end of the previous section. Let us consider the collapse from the rest and put t?(O) = q(O), g&O) = q+(O) = 0. If, in addition, we require that q(0) > qo (or q(0) > q*), the &particle will move towards +oo under the action of the force increasing monotonically. Since qo (and q*) w log( ]Ej/MJ>, this requirement can be funneled by choosing an initial state with a finite amplitude, but with a sufficiently small average energy density. The equation of motion of the q-particle in the potential (22) can be exactly integrated in an explicit form only for several values of s, for example, for s 2 (cb4 theory). The result is expressed in terms of elfiptic functions. To obtain an estimate valid for any f we consider two cases. Suppose first that q(0) > qo (or q( 0) z+ q*). This case corresponds to the collapse of states having large amplitudes, i.e. with A(0) > (B/p) ~I~~/~~)‘/~. Hence, we can integrate the equation for the ~-particle neglecting the first term in (22). This gives
A(t) For 3(x) increasing at infinity faster than x log x the integral (20) converges, so that in this case the collapse occurs in a finite time.
_ 4hP&2/n e(~- 1)(I
(21)
This estimate is valid for 0 -=ct < t, < &. It becomes exact when E = 0. For q(O) arbitrary the result can be obtained near the moment of the collapse, i.e. when 1 - t/t, < 1,
EM
50
Maslov. A.G. Shugalav/Physics Leners A 239 (1998) 4tS.W
under the additional assumption that 1 - t,,& << 1. In this case we have
_A(t) >
,6(+4(o)
x
A(0)
const (7, - @/(J-U ’
(25)
where const = [Z(s - 1)2A~]-‘/fs-1)~/~A(0), while tC = i21&c is calculated from (20), (22). For s = 2 the estimate (25) was given in Ref. [ lo]. As a second example we choose U( 4) = A( t - cash #)
F(4*)=A(!-cosh@-t-$@sinh@), (27) which is a convex non-negative function of 4*. The calculation of the effective potential yields =
-&eeq
+ 2~~~‘~ f 1 + 2 em4( 1 - cash eq”) 1.
(28)
Near the collapse the equation for the +particle can be integrated explicitly. This gives
-A(t) > e4frf-4fo)M const A(O)
Acknowledgement The authors thank the Russian Foundation for Basic Research for support through the grant No. 97-0216561.
(26)
(A > 0), which corresponds to the sinh-Gordon equation. This equation describes the dynamics of cosmic strings in constant curvature spacetimes [ 151. Its timeindependent version appears in two-dimensional hydrodynamics and in pIasma physics as a model equation determining the equilibrium density dis~ibutions for “negative-temperature”s~tes of an interacting vortices ensemble or guiding center plasma [ 161. As U( 4) is even, we can write F( 4*) as
U(q)
where const = 4~/~A(O~, while ic is calculated from (20)) (28). The estimate (29) is valid when 1 t/tc << 1 provided 1 - t,/& CC 1.
X IOg*
,
(29)
References 111YE. Zakharov, in: Handbook of Plasma Physics, Vol. 2, eds. A.A. G&e, R.N. Sudan (North-HoRand, Amsterdam, 1984) p. 81. I21 M.V. Goldman, Rev. Mod. Phys. 56 (1984) 709. 131 J.J. Rasmussen, K. Rypdat, Physica Scripta 33 (1986) 481. t41 K. Rypdal. J.J. Rasmussen, Physica Scripta 33 (1984) 498. r51 E.A. Kuznetsov, J.J. Rasmussen, K. Rupdaf, SK. Turitsyn, Physica D 87 (1995) 273. [61 PM. Lushnikov, Pis’ma Zh. Eksp. Teor. Fiz. 62 (1995) 447. r71 E.M. Maslov, A.G. Shagalov, in: Nonlinear evolution equations and dynamical systems, Proc. NEEDS’92, eds. V. M~~kov, 1. Puzynin, 0. Pashaev (World Scientific, Singapore, 1993) p. 159. 181 EM. Maslov, A.G. Shagatov, in: Nonlinear Physics: Theory and Experiment, eds. E. Altinito, M. Boiti, L. Mattina. F. Pempinelli (World Scientific, Singapore, 1996) p. 421. 191 E.M. Maslov, A.G. Shagatov, Phys. Lett. A 224 ( 1997) 277. r 101 EA. Kuznetsov, PM. Lushnikov, Zh. Eksp. Teor. Fiz. 108 (1995) 614. IIll G.H. Hardy, J.E. Littlewood, G. Polya, rn~u~ities (Cambridge Univ. Press, Cambridge, 1951). [I21 J. Szarski, Differential lnequahties (Warsaw, 1965) Ch. 4. 1131 A.R. Bishop, I.A. Krumhansl, S.E. Trullinger. Physica D 1 (1980) 1. iI41 A. Linde, Particle Physics and Cosmology (Gordon and Breach, London, 1990). 1151 A.L. Larsen, N. S&nchez, Phys. Rev. D 54 ( 1996) 280 1. [I61 D. Montgomery, G. Joyce, Phys. Fluids 17 (1974) 1139.
PHYSICS
ELSEMER
LETTERS
A
Physics Letters A 239 ( 1998) 46-50
On the dynamics of scalar wave coiiapse E.M. Maslov **I,A.G. Shagalov bs a fmtitu~eof Terrestrial ~ag~edsm,
ionosphere and Rad~owave Propagafion of the Russian Academy of Sciences (IZM~~N~, Troifsk, Moscow Region 142092. Russia b institute of Metal Physics of the Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya 18, GSP-170, Ekaterinburg, 620219, Russia
Received 5 November 1997; accepted for publication 2 December 1997 Communicated by V.M. Agranovich
Abstract The collapse in the nonlinear Klein~ordon equation is considered in a general way for a sufficiently wide class of nonline~ties. A ~ffe~ntial in~u~ity making it possible to estimate the collapse dynamics is derived. As examples, the estimates are obtained for some nonline~ties having physical interest. @ 1998 Published by Elsevier Science B.V. Keywords: Multidimensional solitons; Collapse
1. Introduction The explosive increase of a field in a finite space region is a widespread manifestation of instability in nonlinear dynamical systems. In the physics literature this phenomenon is usually referred to as wave collapse, since in most cases the increase of a field amplitude is accompanied by a contraction of a field distribution. The corresponding dynamical structures (frequently called collapsons, explosons, etc.) are essentially nonlinear and, as a rule, become singular in a finite time. The best-investigated dynamical systems in which the collapse phenomenon occurs are the Zakharov equatians and the nonlinear Schrtidinger equation with a power nonlinearity. In these cases a number of both numerical and self-similar solutions describing a field near singul~ity have been found (see, e.g., ’ Corresponding author. E-mail: [email protected] 2 E-mail: arkadi.shagaiov9usu.m
Refs. [ l-41 ), as well as the sufficient conditions for the initial field distributions under which the collapse occurs have been derived [ $6 ] _ In the relativistic case, in particular, for rhe nonlinear Klein-Gordon equation, the collapse is less well understood. In Refs. [7-91 we have obtained the sufficient conditions for the collapse in the nonlinear Klein-Gordon equation with a logarithmic nonlinearity and have also investigated numerically the collapse dynamics. The corresponding sufficient conditions in the c$~model have been found in Ref. [ IO]. In the present Letter we make an attempt to investigate in a general way the collapse in the real nonlinear Klein-Gordon equation, &t - W -t lJ’ (4) = 0,
(1)
for a sufficiently wide class of nonline~ities. Namely, we only suppose that U(4) - $4 U’ (#) is a convex non-negative differentiable function of tb2. It turns out that for such potentials the approach developed in
037%9601/98/$19.00 @ 1998 Published by Elsevier Science B.V. All rights reserved. PfI SO375-9601(97)00958-4
E.M. Maslov, A.G. Shagalov/Physics
Refs. [7-91 can be generalized giving a useful tool for investigation of dynamics of the collapse. As an illustration, we obtain some estimates for collapse dynamics in the model with a polynomial nonlinearity and in the sinh-Gordon model.
Letters A 239 fl998)
41
46-50
Taking into account the Cauchy-Schwarz inequality A: Q 4AB, from (7) we obtain the nonlinear differential inequality A,t - A;/A
- 4
JF(@‘)
dr i- 4E 3 0
(8)
W” 2. The basic inequality
supplemented with the initial conditions
Let us consider a lump of a real scalar field possessing a finite energy
A(0)
E=
[~(#t)2+~(V~)2+U(~)]dr
s ‘F?’
(the integration is over space IX”;the field 4 (r, the potential U {$) are ficiently fast at infinity). lump as sb(r,O)=f(r),
(2)
the n-dimensional Euclidean t) , its spatial derivatives, and assumed to be vanishing sufDenote the initial state of the
SSt(r,O)=g(r).
(3)
These two functions determine the value of the energy E. On the other hand, by analogy with Refs. [ 7-91, we can recast Eq. (2) into
where F(#2) = U(4) - $$ W$)
(5)
is assumed to be a function of #2. Notice, that F(0) = 0. Considering E as a known quantity and denoting A(t) = /42(r,t)dr. W”
B(t)
=
J&(r, R"
t) dr
J
F(+2)dr=4(B-E).
118”
f”dr,
A,(O) = 2
J
fgdr.
(9)
W”
This inequality is the generalization of the one obtained in Ref. [7]. The function F(4*) is determined by the potential U( 4). Conversely, for a given F( +2) expression (5) can be regarded as a differential equation determining U(4). Let, for example, F( (b2) = ( A/2)#2 (A > 0). Then from Eq. (5) one finds U(#) = -(A/2)+2 log @2 + const x $2. This case has been investigated in detail in Refs. [ 7-9]. In particular, the conditions for initial data (3) have been found there under which the field amplitude squared begins to increase extremely fast (N exp( At2) ) in a finite space domain (N h-‘/2). In the general case one fails to find conditions of this type. The evident exception is the states with E < 0. As is seen from (8)) in this case for any F(#‘) 3 0 any initiat data lead to the unlimited increase of A(t), in agreement with the results of Refs. [ 7-91. In the present Letter we proceed from the assumption that the collapse takes place a priori, and that it is evolving in a finite space domain. To investigate its dynamics we invoke the following simple physical considerations. Define a center and a radius squared of the collapsing lump respectively as
(6)
1 A
JrqS2 dr,
W”
{Ar2>= ;
J (r- al242dr,
( 10)
HP”
we rewrite (4) in the form Al,-4
J
W”
a=-
J
=
(7)
where A is defined in (6). Suppose that (Ar2)‘/2 as a function of time has a least upper bound R, R = sup,(Ar 2) ‘I2 . Physically, this means that at all times the increase of the field amplitude occurs mainly in
48
E.M. Maslov, A.G. Shagalov/Physics
the ~-dimensional ball ll8: of the fixed finite radius R centered at a. Define a function m(t) as m=
-&&ti2dr.
Letters A 239 (1998) 46-N
that for the weII-localized collapsing states with sufficiently large amplitude the values of R are not too large (of the order of half the initial lump), while ,u is close to the unity. 3. The effective potential and comparison
From the definition of B;t, it is clear that for the collapsing states 0 < ,u 6 m(t) < 1, where fi = inf, m. Thus we can write
theorems
With the help of the transformation eq(r) = @2-‘A(t),
r z &?-‘/“t,
(12) I = 4@+-‘E, Let now F( 42) (5) be a non-negative convex differentiable function of 4*. Since F(0) = 0, this implies that F(42) and its derivative with respect to +2 increase monotonically with increasing +*. On the other hand, because F‘(42) is convex, the Jensen inequality is valid (see, e.g., Ref. [ 111). Thus, taking into account (12) we have
< $
qJ
F(42)dr
< ;
IT+*) dr, JW”
(13)
3(eq)
= 4,~&*f”F(,u~~‘A)
(16)
inequality (15) takes the Newtonian form qn > -m/dq
(17)
with the effective potential
U(q) = 4
3( e*) e-* dq.
e-4 J
(18)
Consider the properties of the potential (18) taking into account the above suppositions concerning F(t$2). In accordance with them 3(eq) is a convex non-negative differentiable function of e4 increasing monotonically, with 3(O) = 0. Therefore, from ECq.( lS> we have (omitting an arbitrary constant)
where
n=
J
dr=
.rm/* R”.
l-(42
+ 1)
U(q) --+ -Eew4 - 3’(O) q - %3”(O) e4 - . . . (14)
(4 --) -co)9 U(q)/q
Returning to (8) with this result we obtain A,, - A;/A
c 4E > 4f2F( (p/%)A).
(151
This inequality supplemented with the initial conditions (9) serves as a basis for the subsequent analysis. We call attention to the following. The r.h.s. of ( 15) contains two unknown constants: 0 < p < 1 and 0 < R < 00. Their exact values can be calculated only knowing the solution +(t, t) for all t > 0. This prevents the finding of the sufficient conditions for the coIlapse in the spirit of Refs. [ 7-91. Nevertheless, if the collapse takes place, its dynamics can be investigated on the basis of the inequality (15) supposing only the existence of the above constants. Notice
--, --oo
(4 --+ +cQ)T
(1%
where primes denote the derivatives with respect to x = ee. The corresponding “force” in the r.h.s. of ( 17) has the form -d U/dq = e-9[ 3( eq) -&I . It is important that for & 2 0 this force increases monotonically with increasing q. Indeed, taking the derivative with respect to q one has -d2U/dq2 = e-q[ l+eq3’( eQ) 3( eq) J . This expression is positive, because any convex function 3(x) such that 3(O) = 0 satisfies the inequality x3’(x) > 3 (n) (x > 0). For & > 0 the force is zero at q = qo, where 3(eqo) = 8. At this point the potential has a maximum. For & < 0 the regions of monotonic variation of the force are separated by a point q* de~rmined From the equation e**3’(e@ ) - 3( eq*) = -&. The force decreases
where s > 1 is an integer, A > 0. This potential belongs to the class of polynomial ones. Such potentials occur, for example, in the theory of water wave generation by wind [ lo], and, for negative A, in condensed matter physics [ 131, in particle physics and cosmology [ 141 fd4 theory). With the potential (2 I f we have F(#?)
= h(s - 1)&S,
and hence 3(e9> =4ff - i)A&Pe4”. From ( 18) we thus obtain Fig. I. The qualitative behavior of the effective potential ( 18) for various energies of the collapsing lump.
when --00 < 4 < q, and increases when q* < 4 < co with increasing q. At 4 = qr the corresponding curve U(g) has an inflection point. So, we have generally the picture presented in Fig. 1. In the collapse A - eq tends to -+-co. To obtain estimates for its rate let us consider the motion of a Newtonian particle. say Q. obeying, instead of ( 17), the equation & = -d U( @ ,/d& Suppose the initial conditions (e.g., for T = 0) are so chosen that the 4particle moves in the region of monotonic increase of the force. Then, in accordance with the comparison theorems f 121, the inequality q(7) > &T) is valid for 0 < T < rc provided the initial conditions for the g-particle. and the &particle coincide. Obviously, the collapse time 7-Ccan be estimated as
U(q) = -&e-q
A0
>
e8(7Mo)
(23)
where
(24)
4. Examples As a first example we take U(ss) = -,c$~” + const x &,
(22)
Now we use the considerations given at the end of the previous section. Let us consider the collapse from the rest and put t?(O) = q(O), g&O) = q+(O) = 0. If, in addition, we require that q(0) > qo (or q(0) > q*), the &particle will move towards +oo under the action of the force increasing monotonically. Since qo (and q*) w log( ]Ej/MJ>, this requirement can be funneled by choosing an initial state with a finite amplitude, but with a sufficiently small average energy density. The equation of motion of the q-particle in the potential (22) can be exactly integrated in an explicit form only for several values of s, for example, for s 2 (cb4 theory). The result is expressed in terms of elfiptic functions. To obtain an estimate valid for any f we consider two cases. Suppose first that q(0) > qo (or q( 0) z+ q*). This case corresponds to the collapse of states having large amplitudes, i.e. with A(0) > (B/p) ~I~~/~~)‘/~. Hence, we can integrate the equation for the ~-particle neglecting the first term in (22). This gives
A(t) For 3(x) increasing at infinity faster than x log x the integral (20) converges, so that in this case the collapse occurs in a finite time.
_ 4hP&2/n e(~- 1)(I
(21)
This estimate is valid for 0 -=ct < t, < &. It becomes exact when E = 0. For q(O) arbitrary the result can be obtained near the moment of the collapse, i.e. when 1 - t/t, < 1,
EM
50
Maslov. A.G. Shugalav/Physics Leners A 239 (1998) 4tS.W
under the additional assumption that 1 - t,,& << 1. In this case we have
_A(t) >
,6(+4(o)
x
A(0)
const (7, - @/(J-U ’
(25)
where const = [Z(s - 1)2A~]-‘/fs-1)~/~A(0), while tC = i21&c is calculated from (20), (22). For s = 2 the estimate (25) was given in Ref. [ lo]. As a second example we choose U( 4) = A( t - cash #)
F(4*)=A(!-cosh@-t-$@sinh@), (27) which is a convex non-negative function of 4*. The calculation of the effective potential yields =
-&eeq
+ 2~~~‘~ f 1 + 2 em4( 1 - cash eq”) 1.
(28)
Near the collapse the equation for the +particle can be integrated explicitly. This gives
-A(t) > e4frf-4fo)M const A(O)
Acknowledgement The authors thank the Russian Foundation for Basic Research for support through the grant No. 97-0216561.
(26)
(A > 0), which corresponds to the sinh-Gordon equation. This equation describes the dynamics of cosmic strings in constant curvature spacetimes [ 151. Its timeindependent version appears in two-dimensional hydrodynamics and in pIasma physics as a model equation determining the equilibrium density dis~ibutions for “negative-temperature”s~tes of an interacting vortices ensemble or guiding center plasma [ 161. As U( 4) is even, we can write F( 4*) as
U(q)
where const = 4~/~A(O~, while ic is calculated from (20)) (28). The estimate (29) is valid when 1 t/tc << 1 provided 1 - t,/& CC 1.
X IOg*
,
(29)
References 111YE. Zakharov, in: Handbook of Plasma Physics, Vol. 2, eds. A.A. G&e, R.N. Sudan (North-HoRand, Amsterdam, 1984) p. 81. I21 M.V. Goldman, Rev. Mod. Phys. 56 (1984) 709. 131 J.J. Rasmussen, K. Rypdat, Physica Scripta 33 (1986) 481. t41 K. Rypdal. J.J. Rasmussen, Physica Scripta 33 (1984) 498. r51 E.A. Kuznetsov, J.J. Rasmussen, K. Rupdaf, SK. Turitsyn, Physica D 87 (1995) 273. [61 PM. Lushnikov, Pis’ma Zh. Eksp. Teor. Fiz. 62 (1995) 447. r71 E.M. Maslov, A.G. Shagalov, in: Nonlinear evolution equations and dynamical systems, Proc. NEEDS’92, eds. V. M~~kov, 1. Puzynin, 0. Pashaev (World Scientific, Singapore, 1993) p. 159. 181 EM. Maslov, A.G. Shagatov, in: Nonlinear Physics: Theory and Experiment, eds. E. Altinito, M. Boiti, L. Mattina. F. Pempinelli (World Scientific, Singapore, 1996) p. 421. 191 E.M. Maslov, A.G. Shagatov, Phys. Lett. A 224 ( 1997) 277. r 101 EA. Kuznetsov, PM. Lushnikov, Zh. Eksp. Teor. Fiz. 108 (1995) 614. IIll G.H. Hardy, J.E. Littlewood, G. Polya, rn~u~ities (Cambridge Univ. Press, Cambridge, 1951). [I21 J. Szarski, Differential lnequahties (Warsaw, 1965) Ch. 4. 1131 A.R. Bishop, I.A. Krumhansl, S.E. Trullinger. Physica D 1 (1980) 1. iI41 A. Linde, Particle Physics and Cosmology (Gordon and Breach, London, 1990). 1151 A.L. Larsen, N. S&nchez, Phys. Rev. D 54 ( 1996) 280 1. [I61 D. Montgomery, G. Joyce, Phys. Fluids 17 (1974) 1139.