Dynamics of wave collapse in the critical case

Volume 151, number 6,7

PHYSICS LETFERS A

17 December 1990

Dynamics of wave collapse in the critical case V.M. Malkin Institute ofNuclearPhysics, 630090 Novosibirsk, USSR Received4 December 1989; revised manuscript received 25 June 1990; accepted for publication 18 October 1990 Communicatedby A.P. Fordy

An accurate analytical solution ofthe well-known problem of collapse dynamics for the soliton described by a two-dimensional Schrödinger equation with cubic nonlinearity is found.

The nonlinear Schrödinger equation

(~~ —~f- - ~ +

~-

rd_i

-~-

or

+

I wI 4/d}~~=0

(1)

—

at d= 2 was considered originally as a mathematical model ofthe self-focusing phenomenon that had been predicted in ref. [1] for powerful radiation flows in a medium [2—5] (see also reviews [6,7]). Later on this model turned out to be applicable as well to the surface waves in a deep liquid [81 (see also review [9]). The voluminous literature on eq. (1) now available contains a number ofimportant results (see, for example, review [10]). At an early stage of the investigations of eq. (1) the possibility of explosive self-compression “collapse” of wave packets was —

investigations the nontrivial symmetry [11] of eq. (1) was of significant importance. This symmetry enables Talanov to construct a collapsing soliton whose size decreases in time by the law a t and

—

the form coincides exactly with that ofthe stationary soliton. The instability of such a solution is evident from the very fact that the number of quanta in the soliton is exactly equal to the critical one. Later on numerous attempts to find stable collapsing solutions were undertaken on the basis of asymptotic generalizations of the Talanov transformation [11, 12]. The following formulae were suggested and tested numerically at different stages of the investigations, a—’ (tm— )2/3 [13—16], a (t~—t)1/2 [17,18],

( ~

1/2

revealed and the necessary condition for such a blowup was formulated:

a

N= Jdrrd_iIWI2>Nc.

a~( t~—t )i/2 ln{1n[1/(t~—t)]}

The critical value N~coincides with the “number of quanta” N in the stationary soliton: (2) W(r,t)=a”2R(r/a)exp(it/a2),

The latter are from refs. [19—23]and refs. [24—26]

(

1+

R (~)

—~

‘d~

+ IR(~)I4/d)R(~)=0, d~

0.

(3)

This number is independent of the soliton size a and is equal to 1.86 at d=2 (see ref. [2]). For further Elsevier SciencePublishers B.V. (North-Holland)

—

In [1/(t~

~ —

t)])

respectively. It is noteworthy that the law a=(21t ln{ln[l/(t*_t)]}) 1/2 ____________

(4)

follows immediately from the formulas (5.11)(5.13) and definitions (3.1), (3.2) of ref. [24]. (The presence of a single logarithm rather than a double one in the concluding formula of ref. [24] was caused by a slip of Friman’s pen.) However, the proofofthe key formula (5.11) in ref. [24] is not 285

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satisfactory and this was clearly understood by Friman himself. In fact, in the conclusion of ref. [24] he wrote: we failed to find a mechanism, more effective than that described in §5, for the parametric transfer of quanta from the peak to the background directly in the region of the peak. The possibility for such a mechanism to exist remains open”. Note that Friman had discussed three mechanisms for the transfer of quanta from the “peak” (i.e. the soliton of form (3)) to the background. The probabilities for nonadiabatic transition and tunneling decrease exponentially when the parameter i~. (see ref. [24]) tends to infinity, while the parametric interaction between the peak and the background decreases only as an inverse power of i~. Nevertheless, Friman had assumed that the main mechanism for losing quanta by the peak is tunneling, In refs. [25,26] Papanicolaouand co-workers have rediscovered Friman’s result. The main idea or refs. [25,261 is to approximate the solution of the nonlinear Schrödinger equation in the critical case by self-similar solutions existing in slightly supercritical cases. The supercriticality depends on some parameter (similar to Friman’s ~2) and tends to zero when ~2 —p0. The asymptotics of this dependence should have been found analytically from the “nonlinear eigenvalue problem” (see eq. (27) of ref. [26]), but the authors gave only a rough intuitive estimate and based their further treatment on the extrapolation of numerical results obtained at not very small supercriticalities. The stability problem for the given self-similar solutions was not considered in refs. [25,26] at all, though the authors had determined the time dependence of the zero-order solution in the same way as Friman, whose result was based, however, on the well-known stability properties of the soliton. Note also that the supercritical self-similar solution contains an infinite number of quanta, in contrast to the exact critical solution. Consequently, the relation between the zero-order and the exact solu“...

tions is not so simple as Pananicolaou and co-workers assumed. On the base of their numerical calculations [22,25,26] the authors “conclude that the correction to the (I,. t) 1/2 scaling factor is a function which varies more slowly than a logarithmic power”. However, as advanced as in ref. [251 the corn—

286

17 December 1990

putation, undertaken by Zakharov and Shvets (see ref. [23]), allows quite another conclusion: the authors claimed that the power y ( t) in the conventional law 2

a(t)=constx

(t~—t)~

is restricted by inequalities 0.35 ~ y( t~)~ 0.65. In order to find y ( t~)Zakharov and Shvets formulated in ref. [27] the nonstationary nonlinear eigenvalue problem, but did not solve it. In the present paper a generalized Talanov transformation is applied in such a form that the additional potential (see the term proportional to the p2 in eq. (10) below) turns out to be time-independent. A proper cutting ofthe additional potential ~2, say at p = 1, leads to the modified soliton, which is already “dressed” by the parametric interaction with the background (see eq. (14) below). Such a renormalization suppresses the background compared with the Friman approach and allows one to distinguish exponentially small effects. In contrast to the self-similar solution, used by Papanicolaou and coworkers, the modified soliton contains a finite number of quanta. When this number is given the modified soliton, contrary to the “naked” one, is rigidly fixed in size. The exact solution of eq. (10) “glides” along the set of the modified solitons and the only driving force of this “gliding” is the tunneling. The smallness of the tunneling-induced parametric and nonadiabatic effects is easily checked and implies the seif-consistency of the solution. This draft is formalized mathematically below. The substitution t

w( r, 1) =f(p, t) argf(0, r) = 0,

exp(~dr+i

J

r=ln[ l/(t~

))~

dt~~c(r~

—

I)],

p= re~’2

reduces eq. (1) to

[~(~

+ ~d+ iK + ~p

+ If I 4/d] f= 0

.

+

~

~

pd_i

(5)

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Then it becomes clear that at ‘r-~ cx~the value of K also tends to infinity, though much slower than t (as ln r). So, being interested in the solution’s asymptotics at l—~t~, one can assume K>> 1. In the range p ~ ‘/2 all the terms of eq. (5) are small compared with the first one and the general solution is given by the formula f(p, r) =F(r—2 ln p)

J

Xexp(_~dlnp_i

(6)

dTiK(ri))

(8), (9). The standard quasi-classical calculation (see, for example ref. [281) gives ~c(r)=

—

—lnIF(r)I2,

2it arg F(r)=ic(r) [In K(r)+2 in 2—11 + ~

(13)

~

p

0

corresponding to the “frozen” field beyond the soliton: 1) =F( —2

ln r) / —2~~nr Xexp(~—~dlnr+i dtK(T)).

J

(7)

(12)

The time dependences are to be calculated with the help of the exact relationship

r—2Inp

W(r,

17 December 1990

In the range In K <
At not very large values of p formula (6) is simplifled significantly: f(p, r)=F(~r)p

/

I d/2+2i,c(~)J

~

,

In,c<
l+_~):d_i~_+ ed—i

d~

(8)

(9)

f(p,r)=J(p,r)exp(—~ip2), /

\

(i-~-—K+—~—-~-

\,

&r

pd_i

0~

16P

j

‘

(10) 2the solution of eq. (10) coinIn thewith range p<<~’/ cides the stationary soliton (3) with an accuracy up to a term small over the parameter K ‘: J(p’

“

K’~ 4/d /

-~—~9(K—~5) JS(~,K)=0, /

S(~,K)

—.

ø(x)0, =1,

0,

(14)

x<0, x>0.

21J.0

d-I°~Ij~4/d+±

0~,P

~

~2

+

Starting to solve eq. (5) in the internal range p ~ one should separate the spatial oscillations from the functionf(p, r):

d~

Since S(~,oo)=R(i~),the number of quanta N(K) at K>> 1 can be calculated with the theory of perturbations: Al N(K) =N~+ —i, M=~ d~’~IR(~)12. (15)

J

K

0

(11)

Putting N(ic) into the left hand side ofeq. (13) and

In the range p>>K~’2 the function Ill is already small and the nonlinear term in (10) can be neglected. At lnp<< r the term with 0/Or is also negligible. The linear stationary equation can be solved analytically. The solution decreases exponentially at K 112<

> K1 ~‘2 goes to the asymptoteJ’~exp ( ~ip~)in agreement with formulae

formulae (8), (9) into its right hand side and taking into account (12) yields 2M I F( r) 12 = ~ K( r) = i— ln ‘r. (16)

—

t)

~K~4R(K”2p)

.

Going back to the original variables, one can easily obtain the law (4). The field w( r, t~)near the singularity at the moment of its formation is to be cal287

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culated by formulae (7), (16). The singularity of the quantum density at r—~0turns out to be integrable (as was bound to happen):

I W(r,

t~)12

8~2M[ln(l/r)]~{ln[ln(l/r)]}3. rd

(17)

The analysis shows that a small perturbation of the initial conditions leads only to a shift of the moment of singularity formation t,~,but does not change the formulae obtained. The author thanks V.F. Shvets for the remark on refs. [25—27].

References [I] G.A. Askar’jan, Zh. Eksp. Teor. Fiz. 42 (1962)1567. [2] R.Y. Chiao, E. Garmire and C.H. Townes, Phys. Rev. Lett. 13 (1964) 479. [3] V.1. Talanov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7 (1964) 564. [4] VI. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 2 (1965) 218. [5] P.L. Kelley, Phys. Rev. Lett. 15 (1965) 1005. [6] V.N. Lugovoi and A.M. Prokhorov, Usp. Fiz. Nauk Ill (1973)203. [7] G.A. Askar’jan, Usp. Fiz. Nauk 111(1973) 249. 18] V.E. Zakharov, PMTF, No. 2 (1968) 86.

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[9] HG. Yuen and B.M. Lake, in: Solitons in action (Academic Press, NewYork, 1978) ch. 5. 110] J.J. Rasmussen and K. Rypdal, Phys. Scr. 33 (1986) 481. [11] VI. Talanov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9 (1966) 410. 112] V.1. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 11(1970) 303. [131 V.E. Zakharov, V.V. Sobolev and V.S. Synakh, Pis’ma Zh. Eksp.Teor.Fiz. 14(1971)564. [14]V.E. Zakharov and V.S. Synakh, Zh. Eksp. Teor. Fiz. 68 (1975) 940. [15] K. Konno and H. Suzuki, Phys. Scr. 20 (1979) 382. [16] P.L. Sulem, S. Sulem and A. Patera, Commun. Pure Appi. Math. 37 (1984) 755. [17] L.M. Degtjarev and VI. Krylov, Zh. Vychisl. Mat. Mat. Fiz. 17 (1977) 523; DokI. Akad. Nauk SSSR 241 (1978) 64. [18]K. Rypdal, J.J. Rasmussen and K. Thomsen, Physica D 16 (1985) 339. [191SN. Vlasov, L.V. Piskunova and VI. Talanov, Zh. Eksp. Teor.Fiz.75 (1978) 1602. [20] D. Wood, Stud. Appl. Math. 71(1984)103. [21] K. Rypdal and J.J. Rasmussen, Phys. Scr. 33 (1986) 498. [221 D.W. McLaughlin, G.C. Papanicolaou, C. Sulem and P.L. Sulem, Phys. Rev. A 34 (1986) 1200. [23] V.E. Zakharov and V.F. Shvets, Pis’ma Zh. Eksp. Teor. Fiz. 47 (1988) 227. [24] G.M. Friman, Zh. Eksp. Teor. Fiz. 88 (1985) 390. [25] B.J. Le Mesurier, G.C. Papanicolaou, C. Sulem and P.L. Sulem, Physica D 31(1988) 78; 32 (1988) 210. [261MG. Landman, G.C. Papanicolaou, C. Sulem and P.L. Sulem, Phys. Rev. A 38 (1988) 3837. [27] V.E. Zakharov and V.F. Shvets, Toward a theory of critical wave collapse, in: Proc. KievInt. Workshop on the Nonlinear World (1989), to bepublished. [281 L.D. Landau and E.M. Lifshits, Kvantovaja mekhanika (Nauka, Moscow, 1974) §50.