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New stationary field distributions in nonlinear optics and mechanics of continuous media
Volume 138, number 8
PHYSICS LETTERS A
10 July 1989
NEW STATIONARY FIELD DISTRIBUTIONS IN NONLINEAR OPTICS AND MECHANICS OF CONTINUOUS MEDIA G.L. ALFIMOV, V.M. ELEONSKY, N.E. KULAGIN, L.M. LEHRMANN and V.P. SILIN P.N. Lebedev Physical Institute ofthe Academy ofSciences of the USSR, Leninsky pr. 53, Moscow, USSR Received 19 April 1989; accepted for publication 28 April 1989 Communicated by V.M. Agranovich
New self-localized solutions of the equation A yi— çe+ ~t/~= 0 are given which correspond to (a) periodic chains of the plane nonlinear vortex-type structures of an incompressible fluid, (b) self-localized structures of the laser beam filamentation with complicated symmetry, and (c) the spiral filamentation structures.
1. To describe numerous physical problems one needs to know spatially self-localized solutions of the nonlinear field equation ~w= (1) ~‘—
~
Among them there is the problem of the spatial distribution of the field of electromagnetic beams in nonlinear media, which has been investigated during many years (see for example ref. [11). An older problem is that ofself-localized stationary vortex-type structures in two-dimensional fluids of an incompressible liquid (see for example ref. [21). The selflocalized solutions of eq. (1) with simple (linear or circular) symmetries are well known. We have less information about self-localized solutions with more complicated symmetry. Moreover, it was conjectured in ref. [3] that these solutions do not exist. In this communication we would like to state a new approach, which enables us to obtain a self-localized solution of eq. (1), keeping in mind some applied problems of nonlinear optics, mechanics of continuous media and other branches of physics. The cornplicated symmetry of these solutions indicates the variety of physical properties of systems described by eq. (1). We shall use the results ofref. [4], where the phenomenon of branching-off, corresponding to the birth of new solutions with complicated symmetry near solutions with a simple one was found. In ref. [4] this result was obtained in terms of asymptotic expansions and was associated with the
case under consideration. In the present paper we state that for a number of problems with different symmetry the nonlinear equation (1) permits a branching-off of solutions. The establishing of this fact allows us to formulate an approach to find a solution of eq. (1) and to construct new solutions, which differ from known ones by symmetry (for some preliminary results see ref. [51). 2. The problem is to find solutions of eq. (1) which are self-localized in one variable and periodic in another variable, lim x-. ±=
~(x,y)=0,
~i(x,y+L)=yi(x,y)
.
(2)
Two classes of chain-type solutions that appear are shown in figs. la and lb. The first class includes nonalternating solutions ~(x, y, L). It can be pointed out that as L = 2~t/~J~ these solutions degenerate to the well-known one~dimensionalsolution u(x)= ~J~/ch (x) and when L ~ 1 these solutions form a periodic chain (see fig. la). As L—~cx every “link” of this chain approaches the ground (mode-free) solution u ~ (r) ofthe problem with circular symmetry, U,-,- +
r
—
3
U,- — U + U
r2=x2+y2,
0,
Ur(0)Z0,
urn u(r)=0.
(3)
The second class of solutions is characterized by a
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
443
Volume 138, number 8
PHYSICS LETTERS A
a
10 July 1989
~ /
H
~
/
0 4,~ I
-~c ~
N~V
Fig. 2. Lines oflevels of the two-dimensional solution. ~ o.~
b Fig. 1. (a) Lines oflevels ofthe nonalternating solution (L = 4.6). (b) Lines oflevels ofthe alternating solution (L = 27t).
periodic change ofsign. When L>> 1 the corresponding chain (see fig. lb) consists of links, which apprOximates ±U (0) (r) as L—*oo. In contrast to the first class the period L can decrease to zero. It is accompanied by growth of amplitude and diminishing of the diametrical scale of self-localization. It should be pointed out that the first class of 5O~ lutions can be considered as a set which branches off the solution u (x) = \/~/ ch (x) when L = 2it/ ~ (see ref. [5] for details). A rigorous proof of branchingoff [61 was carried out by means of a method commonly used when one has to prove the existence of periodic solutions in Lyapunov systems. 3. The problem ofthe solutions ofeq. (1) in polar coordinates, (4)
with the dependence on angular variable ~ (i.e. ~ 0) and the following boundary conditions, iirnçt’(r,~)=0, lim~,-(r,~)=0, r—.0
444
(5)
is considered. A numerical algorithm based on the branching-off of the solutions of the nonlinear boundary problem allows us to construct a solution of the problem (4), (5) which possesses a comparatively complicated symmetry (see fig. 2). It is a remarkable property of this solution that its Fourier spectrum with respect to the angular variable ~ is represented by few harmonics. In particular, the approximation ~(r,
)=wo(r)+w1(r)cos4~,+w2(r)cos8q
,
(6)
is good for the solution pictured in fig. 2. This property shows the way to construct a solution of the problem (4), (5), which can be based on an approximation of the solution by several Fourier harmonics and an analysis of the corresponding finitedimensional nonlinear system deduced from (1). 4. The problem of self-localized solutions with spiral symmetry w(z, r, ~
v~—~z)~ ~t(r, ~),
defined by ~(r~,-),-+ (~2+~)W~_W+W3O,
(7)
and the boundary conditions (5) is considered. Fig. 3 illustrates the distribution of the field w( z, r, ~) in thepolarplane (r, q,) foroneofthesesolutions (v=3,
Volume 138, number 8
PHYSICS LETTERS A
10 July 1989
termines the relation between the parameters ~,~i and e. The index 1 corresponds to the number of radial solution ut1~(r)from which the solution involved branches off. The condition of this branching-off
Ie= 0
D 1( ~2 0) = 0 corresponds to the dispersion curves of the linear problem described above. The sections of the manifold D, ( ~,~2 ~)= 0 by the planes ‘-‘ = (n integer) presentdetermine the dispersion 2, e)=0 which the step ofcurves spiral D1(n,u dependent on e (the amplitude of the field). Numerical calculations show that the points in phase space (v ~2 e) that corresponds to the solution moves in the plane v = n with u2 decreasing as ~ grows. Moreover, the case was found when this point attains the plane = 0 as ~ remains finite; this case corresponds to a solution of the problem (4), (5), which is represented in fig. 2. It should be noted that we have used also another approach to obtain this solution independently. According to this ap~,,
Fig. 3. Solution with spiral symmetry (v= 3, step of spiral 2itv/~z= 3.23). Two sections by the planes Z= 0 and Z= 2itv /6/1 are represented.
~j2
2it vLu = 3.23) for two values of z which are one sixth of a step of the spiral apart. 5. Let us set forth the essence of our approach to find self-localized solutions on the example of eq. (7). We rewrite (7) for the function
proach we fixed j~2= 0 from the beginning of the calculations. The dispersion equation D,( v, 0, e) = 0
W(r, Ø)=u~(r)+t~P(r,0)
corresponds to this case.
where u “~(r) is a known self-localized solution with circular symmetry (3) which possesses 1 nodes and ~is an arbitrary parameter. Then the following nonlinear eigenvalue problem for the parameters it, p2 can be discussed,
6. Let us list the results about the spiral symmetrica! solutions of eq. (1), which appear owing to branching off the circular-symmetrical solutions u (1) (r), /0, 1, 2. A numerical analysis shows that the birth of the set of solutions ij.i~,.(r, 0) near circular-symmetrical states u ~ (r) is connected with destruction of one of its critical lines {r=ro; dUM(ro)/dz=0}. The branching off of the ground state which has no critical lines is absent. The distribution of the field with one filament on the axis of the spiral and n filaments situated on some distance from the axis of the laser beam appear in the
(rW~)~+ (it2+ =
~)
W
2~P 00!1’+3[u~’~(r)1
—3eut~(r)!P2—e2 W3,
lim ~P(r,çb)=0,
urn ~Pr(r,0)=0,
(8)
I~1i=1 (9) .
0
It should be pointed out that when c is small and it is possible to neglect the right side of eq. (7) the problem becomes linear and the solution can be found in the class f(r) cos 0 and we obtain an eigenvalue problem for the parameters v, it and the functionf(r). The allowed values of these parameters are represented in the plane (v, it2) by certain curves. The points of intersection of these curves with the lines v=n (n integer) determine a step of the spiral. When e is not small the values of the parameters 1), ~ of the nonlinear problem (8), (9) are given by the dispersion equation D,( v, ~2, e) = 0, which de-
case of branching off U’t (r) as the norm N= ~ (r, 0) — u ~ (r) ~,n = 1, 2, 5, grows. Moreover, as the norm N grows, the structure of a field of central filament approximates the ground state with circular symmetry u (0) (r) and other filaments approximate —Ut0~(r). The lines oflevels of section of this structure in the polar plane when n=3, u= 1.11 are shown in fig. 4. In the case of branching off U~2~ (r) the birth of a new set of solutions can be fulfilled in two different ways. This fact corresponds to two critical lines of solution u (2) (r). Two curves of branching-offofthe linear problem correspond to this case in the plane (it2~v). The distribution of a field ...,
445
Volume 138, number 8
PHYSICS LETTERS A
10 July 1989 W~o
Th~
((0/
:
-
~
I
( Fig. 4. Lines of levels of section of the solution g’,~,,(r,0), 0= n~—/IZ with spiral symmetry by the plane Z=0 (solution was obtained by means ofbranching off u (r) ) -
with one filament on the axis of the spiral and n filaments situated at some distance from the axis appears for n = 1, 2 10 by means of destruction of (r). As the external critical line of u~2~ N=I~w,~,,(r, 0)—U~2t(r)~I grows, the central filament approximates the circular-symmetrical state U (1) (r) and other filaments approximate U (0) (r). The lines of levels of the section of this structure in the polar plane Z=const are shown in fig. 5 for n=7, u= 1.15. However, another way is possible to obtain a set of solutions when n = 2, 3, 4. This way is connected with the breaking of the internal critical line of u (2) (r). In this case the distribution of the field with one filament on the axis of the spiral and n filaments situated at a distance from the axis (fig. 3) appears too. But in this case it is necessary to underline that the structures of filaments cannot be approximated either by U i) (r) or by U (0) (r) or any known simple solutions of the problem with circular symmetry. Let us summarize the results obtained. An approach is suggested and realized to find solutions of eq. (1) which describe (1) plane states like vortex-
446
/
Fig. 5. Lines of levels of section of the solution W,~,,(r,0)’ 0= iup— /2Z with spiral symmetry by the plane Z=0 (solution was obtained by means of branching off u(2) (r)).
type chains and (2) filament states with cylindrical and spiral symmetries. The authors are grateful to N.Y. Ostrovskaya for help in the numerical calculations.
References [1] R.Y. Chiao, E. Garmire and C. Townes, Phys. Rev. Lett. 13 (1964) 479. [21 J .K. Batchelor, An introduction to fluid dynamics (Cambridge Univ. Press, Cambridge, 1970). [3]J.Marburger,Opt.Commun.7 (1973) 57. [4] V.M. Eleonsky and V.P. Silin, Soy. Phys. JETP 29 (1969) 317; 30 (1970) 262. [5] G.L. Alfimov, V.M. Eleonsky, N.E. Kulagin, L.M. Lehrman and V.P. Sum, Two-dimensional self-localized solutions of the equation Au — u + u3 = 0, Prepnnt P.N. Lebedev Physical Institute of the Academy of Sciences of the USSR, N238 (1988). [6] G.L. Alfimov, V.M. Eleonsky, N.E. Kulagmn, L.M. Lehrmann and V.P. Sum, in: Short communications of JINR (Solitons in nonintegrable systems: theory and applications), ed. V.G. Machankov (Dubna, 1989), to be published.
PHYSICS LETTERS A
10 July 1989
NEW STATIONARY FIELD DISTRIBUTIONS IN NONLINEAR OPTICS AND MECHANICS OF CONTINUOUS MEDIA G.L. ALFIMOV, V.M. ELEONSKY, N.E. KULAGIN, L.M. LEHRMANN and V.P. SILIN P.N. Lebedev Physical Institute ofthe Academy ofSciences of the USSR, Leninsky pr. 53, Moscow, USSR Received 19 April 1989; accepted for publication 28 April 1989 Communicated by V.M. Agranovich
New self-localized solutions of the equation A yi— çe+ ~t/~= 0 are given which correspond to (a) periodic chains of the plane nonlinear vortex-type structures of an incompressible fluid, (b) self-localized structures of the laser beam filamentation with complicated symmetry, and (c) the spiral filamentation structures.
1. To describe numerous physical problems one needs to know spatially self-localized solutions of the nonlinear field equation ~w= (1) ~‘—
~
Among them there is the problem of the spatial distribution of the field of electromagnetic beams in nonlinear media, which has been investigated during many years (see for example ref. [11). An older problem is that ofself-localized stationary vortex-type structures in two-dimensional fluids of an incompressible liquid (see for example ref. [21). The selflocalized solutions of eq. (1) with simple (linear or circular) symmetries are well known. We have less information about self-localized solutions with more complicated symmetry. Moreover, it was conjectured in ref. [3] that these solutions do not exist. In this communication we would like to state a new approach, which enables us to obtain a self-localized solution of eq. (1), keeping in mind some applied problems of nonlinear optics, mechanics of continuous media and other branches of physics. The cornplicated symmetry of these solutions indicates the variety of physical properties of systems described by eq. (1). We shall use the results ofref. [4], where the phenomenon of branching-off, corresponding to the birth of new solutions with complicated symmetry near solutions with a simple one was found. In ref. [4] this result was obtained in terms of asymptotic expansions and was associated with the
case under consideration. In the present paper we state that for a number of problems with different symmetry the nonlinear equation (1) permits a branching-off of solutions. The establishing of this fact allows us to formulate an approach to find a solution of eq. (1) and to construct new solutions, which differ from known ones by symmetry (for some preliminary results see ref. [51). 2. The problem is to find solutions of eq. (1) which are self-localized in one variable and periodic in another variable, lim x-. ±=
~(x,y)=0,
~i(x,y+L)=yi(x,y)
.
(2)
Two classes of chain-type solutions that appear are shown in figs. la and lb. The first class includes nonalternating solutions ~(x, y, L). It can be pointed out that as L = 2~t/~J~ these solutions degenerate to the well-known one~dimensionalsolution u(x)= ~J~/ch (x) and when L ~ 1 these solutions form a periodic chain (see fig. la). As L—~cx every “link” of this chain approaches the ground (mode-free) solution u ~ (r) ofthe problem with circular symmetry, U,-,- +
r
—
3
U,- — U + U
r2=x2+y2,
0,
Ur(0)Z0,
urn u(r)=0.
(3)
The second class of solutions is characterized by a
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
443
Volume 138, number 8
PHYSICS LETTERS A
a
10 July 1989
~ /
H
~
/
0 4,~ I
-~c ~
N~V
Fig. 2. Lines oflevels of the two-dimensional solution. ~ o.~
b Fig. 1. (a) Lines oflevels ofthe nonalternating solution (L = 4.6). (b) Lines oflevels ofthe alternating solution (L = 27t).
periodic change ofsign. When L>> 1 the corresponding chain (see fig. lb) consists of links, which apprOximates ±U (0) (r) as L—*oo. In contrast to the first class the period L can decrease to zero. It is accompanied by growth of amplitude and diminishing of the diametrical scale of self-localization. It should be pointed out that the first class of 5O~ lutions can be considered as a set which branches off the solution u (x) = \/~/ ch (x) when L = 2it/ ~ (see ref. [5] for details). A rigorous proof of branchingoff [61 was carried out by means of a method commonly used when one has to prove the existence of periodic solutions in Lyapunov systems. 3. The problem ofthe solutions ofeq. (1) in polar coordinates, (4)
with the dependence on angular variable ~ (i.e. ~ 0) and the following boundary conditions, iirnçt’(r,~)=0, lim~,-(r,~)=0, r—.0
444
(5)
is considered. A numerical algorithm based on the branching-off of the solutions of the nonlinear boundary problem allows us to construct a solution of the problem (4), (5) which possesses a comparatively complicated symmetry (see fig. 2). It is a remarkable property of this solution that its Fourier spectrum with respect to the angular variable ~ is represented by few harmonics. In particular, the approximation ~(r,
)=wo(r)+w1(r)cos4~,+w2(r)cos8q
,
(6)
is good for the solution pictured in fig. 2. This property shows the way to construct a solution of the problem (4), (5), which can be based on an approximation of the solution by several Fourier harmonics and an analysis of the corresponding finitedimensional nonlinear system deduced from (1). 4. The problem of self-localized solutions with spiral symmetry w(z, r, ~
v~—~z)~ ~t(r, ~),
defined by ~(r~,-),-+ (~2+~)W~_W+W3O,
(7)
and the boundary conditions (5) is considered. Fig. 3 illustrates the distribution of the field w( z, r, ~) in thepolarplane (r, q,) foroneofthesesolutions (v=3,
Volume 138, number 8
PHYSICS LETTERS A
10 July 1989
termines the relation between the parameters ~,~i and e. The index 1 corresponds to the number of radial solution ut1~(r)from which the solution involved branches off. The condition of this branching-off
Ie= 0
D 1( ~2 0) = 0 corresponds to the dispersion curves of the linear problem described above. The sections of the manifold D, ( ~,~2 ~)= 0 by the planes ‘-‘ = (n integer) presentdetermine the dispersion 2, e)=0 which the step ofcurves spiral D1(n,u dependent on e (the amplitude of the field). Numerical calculations show that the points in phase space (v ~2 e) that corresponds to the solution moves in the plane v = n with u2 decreasing as ~ grows. Moreover, the case was found when this point attains the plane = 0 as ~ remains finite; this case corresponds to a solution of the problem (4), (5), which is represented in fig. 2. It should be noted that we have used also another approach to obtain this solution independently. According to this ap~,,
Fig. 3. Solution with spiral symmetry (v= 3, step of spiral 2itv/~z= 3.23). Two sections by the planes Z= 0 and Z= 2itv /6/1 are represented.
~j2
2it vLu = 3.23) for two values of z which are one sixth of a step of the spiral apart. 5. Let us set forth the essence of our approach to find self-localized solutions on the example of eq. (7). We rewrite (7) for the function
proach we fixed j~2= 0 from the beginning of the calculations. The dispersion equation D,( v, 0, e) = 0
W(r, Ø)=u~(r)+t~P(r,0)
corresponds to this case.
where u “~(r) is a known self-localized solution with circular symmetry (3) which possesses 1 nodes and ~is an arbitrary parameter. Then the following nonlinear eigenvalue problem for the parameters it, p2 can be discussed,
6. Let us list the results about the spiral symmetrica! solutions of eq. (1), which appear owing to branching off the circular-symmetrical solutions u (1) (r), /0, 1, 2. A numerical analysis shows that the birth of the set of solutions ij.i~,.(r, 0) near circular-symmetrical states u ~ (r) is connected with destruction of one of its critical lines {r=ro; dUM(ro)/dz=0}. The branching off of the ground state which has no critical lines is absent. The distribution of the field with one filament on the axis of the spiral and n filaments situated on some distance from the axis of the laser beam appear in the
(rW~)~+ (it2+ =
~)
W
2~P 00!1’+3[u~’~(r)1
—3eut~(r)!P2—e2 W3,
lim ~P(r,çb)=0,
urn ~Pr(r,0)=0,
(8)
I~1i=1 (9) .
0
It should be pointed out that when c is small and it is possible to neglect the right side of eq. (7) the problem becomes linear and the solution can be found in the class f(r) cos 0 and we obtain an eigenvalue problem for the parameters v, it and the functionf(r). The allowed values of these parameters are represented in the plane (v, it2) by certain curves. The points of intersection of these curves with the lines v=n (n integer) determine a step of the spiral. When e is not small the values of the parameters 1), ~ of the nonlinear problem (8), (9) are given by the dispersion equation D,( v, ~2, e) = 0, which de-
case of branching off U’t (r) as the norm N= ~ (r, 0) — u ~ (r) ~,n = 1, 2, 5, grows. Moreover, as the norm N grows, the structure of a field of central filament approximates the ground state with circular symmetry u (0) (r) and other filaments approximate —Ut0~(r). The lines oflevels of section of this structure in the polar plane when n=3, u= 1.11 are shown in fig. 4. In the case of branching off U~2~ (r) the birth of a new set of solutions can be fulfilled in two different ways. This fact corresponds to two critical lines of solution u (2) (r). Two curves of branching-offofthe linear problem correspond to this case in the plane (it2~v). The distribution of a field ...,
445
Volume 138, number 8
PHYSICS LETTERS A
10 July 1989 W~o
Th~
((0/
:
-
~
I
( Fig. 4. Lines of levels of section of the solution g’,~,,(r,0), 0= n~—/IZ with spiral symmetry by the plane Z=0 (solution was obtained by means ofbranching off u (r) ) -
with one filament on the axis of the spiral and n filaments situated at some distance from the axis appears for n = 1, 2 10 by means of destruction of (r). As the external critical line of u~2~ N=I~w,~,,(r, 0)—U~2t(r)~I grows, the central filament approximates the circular-symmetrical state U (1) (r) and other filaments approximate U (0) (r). The lines of levels of the section of this structure in the polar plane Z=const are shown in fig. 5 for n=7, u= 1.15. However, another way is possible to obtain a set of solutions when n = 2, 3, 4. This way is connected with the breaking of the internal critical line of u (2) (r). In this case the distribution of the field with one filament on the axis of the spiral and n filaments situated at a distance from the axis (fig. 3) appears too. But in this case it is necessary to underline that the structures of filaments cannot be approximated either by U i) (r) or by U (0) (r) or any known simple solutions of the problem with circular symmetry. Let us summarize the results obtained. An approach is suggested and realized to find solutions of eq. (1) which describe (1) plane states like vortex-
446
/
Fig. 5. Lines of levels of section of the solution W,~,,(r,0)’ 0= iup— /2Z with spiral symmetry by the plane Z=0 (solution was obtained by means of branching off u(2) (r)).
type chains and (2) filament states with cylindrical and spiral symmetries. The authors are grateful to N.Y. Ostrovskaya for help in the numerical calculations.
References [1] R.Y. Chiao, E. Garmire and C. Townes, Phys. Rev. Lett. 13 (1964) 479. [21 J .K. Batchelor, An introduction to fluid dynamics (Cambridge Univ. Press, Cambridge, 1970). [3]J.Marburger,Opt.Commun.7 (1973) 57. [4] V.M. Eleonsky and V.P. Silin, Soy. Phys. JETP 29 (1969) 317; 30 (1970) 262. [5] G.L. Alfimov, V.M. Eleonsky, N.E. Kulagin, L.M. Lehrman and V.P. Sum, Two-dimensional self-localized solutions of the equation Au — u + u3 = 0, Prepnnt P.N. Lebedev Physical Institute of the Academy of Sciences of the USSR, N238 (1988). [6] G.L. Alfimov, V.M. Eleonsky, N.E. Kulagmn, L.M. Lehrmann and V.P. Sum, in: Short communications of JINR (Solitons in nonintegrable systems: theory and applications), ed. V.G. Machankov (Dubna, 1989), to be published.