Abstract

The free surface of a liquid film of black oil is deformed by heating with a Gaussian laser beam, thus changing the intensity distribution in the reflected and transmitted beams. The time evolution of the caustics of those beams is numerically studied. The topological structure of the caustics abruptly changes at three critical instants of time. Measurement of the divergence of the caustics of the reflected and transmitted beams allows one to deduce the liquid refractive index.

© 1990 Optical Society of America

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References

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  1. L. Landau, E. Lifschitz, Fluid Mechanics (Mir, Moscow, 1971), pp. 296–297.
  2. V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1962), pp. 384–390.
  3. C. Normand, Y. Pomeau, M. G. Velarde, “Convective Instability: A Physicist’s Approach,” Rev. Mod. Phys. 49, 581 (1977).
    [CrossRef]
  4. T. R. Anthony, H. E. Cline, “Surface Rippling Induced by Surface-Tension Gradients During Laser Surface Melting and Alloying,” J. Appl. Phys. 48, 3888 (1977).
    [CrossRef]
  5. G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
    [CrossRef]
  6. J. C. Loulergue, Y. Levy, C. Imbert, “Thermal Imaging System with a Two-Phase Ternary Mixture of Liquids,” Opt. Commun. 45, 149 (1983).
    [CrossRef]
  7. G. Da Costa, “Real-Time Recording of Light Patterns in Heavy Hydrocarbons: a Theoretical Analysis,” Appl. Opt. 19, 3523–3528 (1980).
    [CrossRef] [PubMed]
  8. G. Da Costa, “Self-Focusing of a Gaussian Laser Beam Reflected from a Thermocapillary Liquid Surface,” Phys. Lett. A 80, 320 (1980).
    [CrossRef]
  9. G. Da Costa, “Competition Between Capillary and Gravity Forces in a Viscous Liquid Film Heated by a Gaussian Laser Beam,” J. Phys. Paris 43, 1503 (1982).
  10. J. Calatroni, G. Da Costa, “Interferometric Determination of the Surface Profile of a Liquid Heated by a Laser Beam,” Opt. Commun. 42, 5 (1982).
    [CrossRef]
  11. G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).
  12. G. Da Costa, “All-Optical Light Switch Using Interaction Between Low-Power Light Beams in a Liquid Film,” Opt. Eng. 25, 1058 (1986).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 120–126.
  14. J. W. Bruce, P. J. Giblin, Curves and Singularities (Cambridge U.P., Cambridge, 1984).
  15. T. Poston, I. Stewart, Catastrophe Theory and its Applications (Pitman, London, 1978), pp. 246–267.
  16. V. I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1984), pp. 29–38.

1986

G. Da Costa, “All-Optical Light Switch Using Interaction Between Low-Power Light Beams in a Liquid Film,” Opt. Eng. 25, 1058 (1986).
[CrossRef]

1983

J. C. Loulergue, Y. Levy, C. Imbert, “Thermal Imaging System with a Two-Phase Ternary Mixture of Liquids,” Opt. Commun. 45, 149 (1983).
[CrossRef]

G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).

1982

G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
[CrossRef]

G. Da Costa, “Competition Between Capillary and Gravity Forces in a Viscous Liquid Film Heated by a Gaussian Laser Beam,” J. Phys. Paris 43, 1503 (1982).

J. Calatroni, G. Da Costa, “Interferometric Determination of the Surface Profile of a Liquid Heated by a Laser Beam,” Opt. Commun. 42, 5 (1982).
[CrossRef]

1980

G. Da Costa, “Self-Focusing of a Gaussian Laser Beam Reflected from a Thermocapillary Liquid Surface,” Phys. Lett. A 80, 320 (1980).
[CrossRef]

G. Da Costa, “Real-Time Recording of Light Patterns in Heavy Hydrocarbons: a Theoretical Analysis,” Appl. Opt. 19, 3523–3528 (1980).
[CrossRef] [PubMed]

1977

C. Normand, Y. Pomeau, M. G. Velarde, “Convective Instability: A Physicist’s Approach,” Rev. Mod. Phys. 49, 581 (1977).
[CrossRef]

T. R. Anthony, H. E. Cline, “Surface Rippling Induced by Surface-Tension Gradients During Laser Surface Melting and Alloying,” J. Appl. Phys. 48, 3888 (1977).
[CrossRef]

Anthony, T. R.

T. R. Anthony, H. E. Cline, “Surface Rippling Induced by Surface-Tension Gradients During Laser Surface Melting and Alloying,” J. Appl. Phys. 48, 3888 (1977).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1984), pp. 29–38.

Bentolila, F.

G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 120–126.

Bruce, J. W.

J. W. Bruce, P. J. Giblin, Curves and Singularities (Cambridge U.P., Cambridge, 1984).

Calatroni, J.

J. Calatroni, G. Da Costa, “Interferometric Determination of the Surface Profile of a Liquid Heated by a Laser Beam,” Opt. Commun. 42, 5 (1982).
[CrossRef]

Chernyakov, A. L.

G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
[CrossRef]

Cline, H. E.

T. R. Anthony, H. E. Cline, “Surface Rippling Induced by Surface-Tension Gradients During Laser Surface Melting and Alloying,” J. Appl. Phys. 48, 3888 (1977).
[CrossRef]

Da Costa, G.

G. Da Costa, “All-Optical Light Switch Using Interaction Between Low-Power Light Beams in a Liquid Film,” Opt. Eng. 25, 1058 (1986).
[CrossRef]

G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).

G. Da Costa, “Competition Between Capillary and Gravity Forces in a Viscous Liquid Film Heated by a Gaussian Laser Beam,” J. Phys. Paris 43, 1503 (1982).

J. Calatroni, G. Da Costa, “Interferometric Determination of the Surface Profile of a Liquid Heated by a Laser Beam,” Opt. Commun. 42, 5 (1982).
[CrossRef]

G. Da Costa, “Real-Time Recording of Light Patterns in Heavy Hydrocarbons: a Theoretical Analysis,” Appl. Opt. 19, 3523–3528 (1980).
[CrossRef] [PubMed]

G. Da Costa, “Self-Focusing of a Gaussian Laser Beam Reflected from a Thermocapillary Liquid Surface,” Phys. Lett. A 80, 320 (1980).
[CrossRef]

Galan, H.

G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).

Giblin, P. J.

J. W. Bruce, P. J. Giblin, Curves and Singularities (Cambridge U.P., Cambridge, 1984).

Gladush, G. G.

G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
[CrossRef]

Imbert, C.

J. C. Loulergue, Y. Levy, C. Imbert, “Thermal Imaging System with a Two-Phase Ternary Mixture of Liquids,” Opt. Commun. 45, 149 (1983).
[CrossRef]

Krasitskaya, L. S.

G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
[CrossRef]

Landau, L.

L. Landau, E. Lifschitz, Fluid Mechanics (Mir, Moscow, 1971), pp. 296–297.

Levchenko, E. B.

G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
[CrossRef]

Levich, V. G.

V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1962), pp. 384–390.

Levy, Y.

J. C. Loulergue, Y. Levy, C. Imbert, “Thermal Imaging System with a Two-Phase Ternary Mixture of Liquids,” Opt. Commun. 45, 149 (1983).
[CrossRef]

Lifschitz, E.

L. Landau, E. Lifschitz, Fluid Mechanics (Mir, Moscow, 1971), pp. 296–297.

Loulergue, J. C.

J. C. Loulergue, Y. Levy, C. Imbert, “Thermal Imaging System with a Two-Phase Ternary Mixture of Liquids,” Opt. Commun. 45, 149 (1983).
[CrossRef]

Normand, C.

C. Normand, Y. Pomeau, M. G. Velarde, “Convective Instability: A Physicist’s Approach,” Rev. Mod. Phys. 49, 581 (1977).
[CrossRef]

Pomeau, Y.

C. Normand, Y. Pomeau, M. G. Velarde, “Convective Instability: A Physicist’s Approach,” Rev. Mod. Phys. 49, 581 (1977).
[CrossRef]

Poston, T.

T. Poston, I. Stewart, Catastrophe Theory and its Applications (Pitman, London, 1978), pp. 246–267.

Ruiz, E.

G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).

Stewart, I.

T. Poston, I. Stewart, Catastrophe Theory and its Applications (Pitman, London, 1978), pp. 246–267.

Velarde, M. G.

C. Normand, Y. Pomeau, M. G. Velarde, “Convective Instability: A Physicist’s Approach,” Rev. Mod. Phys. 49, 581 (1977).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 120–126.

Appl. Opt.

J. Appl. Phys.

T. R. Anthony, H. E. Cline, “Surface Rippling Induced by Surface-Tension Gradients During Laser Surface Melting and Alloying,” J. Appl. Phys. 48, 3888 (1977).
[CrossRef]

J. Opt. Paris

G. Da Costa, F. Bentolila, E. Ruiz, H. Galan, “Interactions entre faisceaux lumineux dans un milieu liquide thermocapillaire,” J. Opt. Paris 14, 179 (1983).

J. Phys. Paris

G. Da Costa, “Competition Between Capillary and Gravity Forces in a Viscous Liquid Film Heated by a Gaussian Laser Beam,” J. Phys. Paris 43, 1503 (1982).

Opt. Commun.

J. Calatroni, G. Da Costa, “Interferometric Determination of the Surface Profile of a Liquid Heated by a Laser Beam,” Opt. Commun. 42, 5 (1982).
[CrossRef]

J. C. Loulergue, Y. Levy, C. Imbert, “Thermal Imaging System with a Two-Phase Ternary Mixture of Liquids,” Opt. Commun. 45, 149 (1983).
[CrossRef]

Opt. Eng.

G. Da Costa, “All-Optical Light Switch Using Interaction Between Low-Power Light Beams in a Liquid Film,” Opt. Eng. 25, 1058 (1986).
[CrossRef]

Phys. Lett. A

G. Da Costa, “Self-Focusing of a Gaussian Laser Beam Reflected from a Thermocapillary Liquid Surface,” Phys. Lett. A 80, 320 (1980).
[CrossRef]

Rev. Mod. Phys.

C. Normand, Y. Pomeau, M. G. Velarde, “Convective Instability: A Physicist’s Approach,” Rev. Mod. Phys. 49, 581 (1977).
[CrossRef]

Sov. J. Quantum Electron.

G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, A. L. Chernyakov, “Thermocapillary Convection in a Liquid under the Action of High-Power Laser Radiation,” Sov. J. Quantum Electron. 12, 408 (1982).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 120–126.

J. W. Bruce, P. J. Giblin, Curves and Singularities (Cambridge U.P., Cambridge, 1984).

T. Poston, I. Stewart, Catastrophe Theory and its Applications (Pitman, London, 1978), pp. 246–267.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1984), pp. 29–38.

L. Landau, E. Lifschitz, Fluid Mechanics (Mir, Moscow, 1971), pp. 296–297.

V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1962), pp. 384–390.

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Figures (16)

Fig. 1
Fig. 1

Liquid film (initial thickness y0) heated by a Gaussian laser beam with diameter 2a at the inflection point of the power distribution.

Fig. 2
Fig. 2

Two possible behaviors of the liquid film in the initial heating stage: (a) dilatation (due to density decrease); (b) contraction (due to the surface tension gradient).

Fig. 3
Fig. 3

Top: dimensionless temperature distribution f(β,γ) at three different time instants (1,2,3); bottom: corresponding liquid surface profile. A pit appears at the top of the surface profile when the maximum value of f(β,γ) is greater than the critical value fc.

Fig. 4
Fig. 4

Interferometric records of the time evolution of the surface profile.10 The sequence runs from left to right and then up and down. The initial dilatation stage and the following contraction are appreciated.

Fig. 5
Fig. 5

Three-dimensional perspective of the surface evolution numerically calculated from Eq. (4) in the text.

Fig. 6
Fig. 6

Direct photographic record of the laser beam reflected from the liquid film at four different time instants8: (a) t = 0 when the laser beam is reflected from the initially undeformed surface; (b), (c) the time instant when light reflected at the concave surface top is focused at the observation plane; (d) formation of interference fringes after passage of the focus through the observation plane.

Fig. 7
Fig. 7

Direct photograph of the laser beam reflected from the deformed liquid film for long heating time. The circular fringes are due to interference between light rays coming from different regions of the deformed surface. The bright external ring is directly reflected from the inflection points of the central region of the liquid film.

Fig. 8
Fig. 8

Incident light ray i reflected at point S of the liquid surface profile. The slope at S is the tangent of angle θ. The reflected ray r is tangent to curve E (envelope) at contact point P. Light intensity is a maximum in the neighborhood of the envelope, which is known as the caustic of the family of reflected rays.

Fig. 9
Fig. 9

(a) Caustic of light rays reflected from a Gaussian surface profile y(x) represented as the border of shadowed regions. The different caustic branches are assigned with the same numbers as the corresponding surface regions. (b) The parametric coordinates xR(x),yR(x) of a generic point of the caustic are represented by dashed and solid lines, respectively. (c) The second derivative y″(x) of the surface profile. It is noted that focal points (2,4,6) proceed from surface points where y″(x) is stationary and that asymptotic branches of the caustics come from inflection points (3,5) of the surface.

Fig. 10
Fig. 10

(a) Different wavefronts (1,V,2,3,4) corresponding to the same surface profile. The cusp ridge, which is the loci of the wavefronts regression points, is also a caustic branch. (b) Wavefronts in (a) are sections of the above surface (known as swallowtail) at points (1,V,2,3,4) corresponding to their homonyms in (a).

Fig. 11
Fig. 11

In each figure (a)–(e) is a graph of the surface profile y(x), the second picture is a graph of y″(x), and the third shows the caustic branches corresponding to the profile y(x). Scales are not respected to emphasize the topological properties of each graph. (a)–(e) correspond, respectively, to the time instants (a) t = tf, (b) tf < t < tc, (c) t = tc, (d) tc < t < tv, (e) ttv. The time instants (tf,tc,tv) when the topological structure of the caustics undergoes an abrupt variation are defined in the text.

Fig. 12
Fig. 12

Detailed structure of the central caustic branch of Fig. 11(b).

Fig. 13
Fig. 13

Caustic branches corresponding to the time instant t = tv. This is the only time instant in the evolution of the laser heated surface in which the lateral surface maxima (4,8) give rise to regression points (C,C′) in the caustic.

Fig. 14
Fig. 14

Transmission of a light ray i through the deformed liquid film. The light ray transmitted within the liquid film is t1, while t2 is the emergent ray. It is shown in the text that the envelope of the family of transmitted rays t2 is the same as the envelope of reflected rays, except for a vertical scale factor.

Fig. 15
Fig. 15

Real caustics of the reflected rays are virtual caustics of transmitted rays and vice versa. The above picture represents a virtual focus F2 and two real foci (F1, F 1 ) of the transmitted beam. These foci correspond to the regression points (4,6,8) in Fig. 11(e). The photographic sequence shows the time evolution of the transmitted beam in the observation plane ∑. The circular fringes are due to interference between light rays coming from the point source F2 and the coherent ring source coaxial with the laser beam passing through (F1, F 1 ).

Fig. 16
Fig. 16

Laser beam reflected from the liquid film in Fig. 15 simultaneously recorded (above photograph). The ratio between the diameters of the reflected and transmitted beams at the same (although arbitrary) time instant depends only on the liquid refractive index.

Equations (53)

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T - T 0 = T c · f ( β , γ ) = T c β 2 1 + γ β 2 d θ θ exp θ ,
ρ ( T ) = ρ 0 + ( ρ T ) T = T 0 · ( T - T 0 ) ,
α ( T ) = α 0 + ( α T ) T = T 0 · ( T - T 0 ) .
y = [ y 0 2 ( ρ 0 ρ ) 3 / 4 + 3 ρ g ( α - α 0 ) ] 1 / 2 ,
u = y y 0 = { [ 1 1 + D f ( β , γ ) ] 3 / 4 + 3 A f ( β , γ ) 1 + D f ( β , γ ) } 1 / 2 ,
u = [ w 3 / 4 - C ( w - 1 ) ] 1 / 2 ( D 0 ) ,
w = 1 1 + D f ( β , γ ) , D = T c · ( ρ T ) T = T 0 ρ 0 , A = T c · ( α T ) T = T 0 ρ 0 g y 0 2 , C = 3 A D = 3 ( α T ) T = T 0 g y 0 2 ( ρ T ) T = T 0 .
u γ = d u d w · w γ = 3 - 4 C w 1 / 4 8 u w 1 / 4 · [ - D w 2 f ( β , γ ) α γ ] .
f c = [ ( 4 C 3 ) 4 - 1 ] / D .
m ( x ) = tan ( π / 2 + 2 θ ) = - 1 / tan 2 θ = tan 2 θ - 1 2 tan θ = y 2 ( x ) - 1 2 y ( x ) .
y r - y ( x ) = y 2 ( x - 1 ) 2 y ( x ) ( x r - x ) ,
y R ( x ) - y ( x ) = m ( x ) [ x R ( x ) - x ] .
d y R ( x R ) d x R = y R ( x ) x R ( x ) = m ( x ) .
x R ( x ) = X - y ( x ) y ( x ) ,
y R ( x ) = y ( x ) + 1 - y 2 ( x ) 2 y ( x ) y ( x ) + 1 2 y ( x ) ,
f ( β , γ ) = [ γ + β 2 - 1 2 γ 2 + θ ( γ 3 ) ] exp ( - β 2 ) ,
u = y ( x , t ) y 0 = 1 + γ [ 3 2 ( A - D 4 ) ] exp ( - β 2 ) + γ 2 { 3 2 ( A - D 4 ) β 2 - 1 2 exp ( - β 2 ) + [ D ( 21 64 D - 3 2 A ) - 9 8 ( A - D 4 ) 2 ] exp ( - 2 β 2 ) } + θ ( γ 3 ) .
u = 1 + γ [ 3 2 ( A - D 4 ) ] exp ( - β 2 ) + θ ( γ 2 )
y ( x , t ) y 0 [ 1 + t τ exp ( - β 2 ) ]
t = 8 t 0 3 ( 4 A - D ) = 2 t 0 D ( C - 3 4 ) .
u = 1 - 3 4 A 2 γ 2 exp ( - 2 β 2 ) + θ ( γ 3 )
y ( x , t ) y 0 [ 1 - ( t τ ) 2 exp ( - 2 β 2 ) ]
y ( x ) = y 0 { 1 + K exp [ - ( x a ) 2 ] } ,
x R a = - 2 β 3 1 - 2 β 2 ,
y R a = y 0 a [ 1 + K exp ( - β 2 ) ] - a 4 K y 0 exp ( β 2 ) 1 - 2 β 2 .
d x R d x = y ( x ) y ( x ) y 2 ( x ) ,
d y R d x = [ y 2 ( x ) - 1 ] y ( x ) 2 y 2 ( x ) - y ( x ) 2 y 2 ( x ) ,
R ( x ) = [ 1 + y 2 ( x ) ] 3 / 2 y ( x ) 1 y ( x ) ,
{ [ x R ( x ) - x ] 2 + [ y R ( x ) - y ( x ) ] 2 } 1 / 2 1 2 y ( x ) 1 2 y ( x ) R ( x ) 2 .
( - a 3 2 , 0 , a 3 2 ) .
- a ( 3 2 ) 3 / 2 , 0 , a ( 3 2 ) 3 / 2 ,
y R a - a 2 4 K y 0
y R a a 2 4 K y 0 · e 3 / 2 2
d y w ( x R ) d x w = y w ( x ) x w ( x ) = - 1 m ( x ) .
y w ( x ) = y ( x ) + m ( x ) [ x w ( x ) - x ] + m ( x ) [ x w ( x ) - 1 ] .
x w ( x ) [ m ( x ) + 1 m ( x ) ] + m ( x ) x w ( x ) + y ( x ) - x m ( x ) - m ( x ) = 0 ,
x w 1 + y 4 2 y ( y 2 - 1 ) + x w y ( y 2 + 1 ) 2 y 2 + y 2 + 1 2 y - x y ( y 2 + 1 ) 2 y 2 = 0.
x w - x w y ( x ) y ( x ) = 1 - x y ( x ) y ( x ) .
x w H = B y ( x ) ,
x w ( x ) = x + B y ( x ) .
y w ( x ) = y ( x ) - B 2 [ 1 - y 2 ( x ) ] .
x w ( x ) = x - 2 K B x y 0 a 2 exp [ - ( x / a ) 2 ] y w ( x ) = y 0 { 1 + K exp [ - ( x / a ) 2 ] } - B 2 { 1 - 4 K 2 ( x y 0 a 2 ) 2 exp [ - 2 ( x a ) 2 ] } ,
x R ( x ) = x - a u u ,
y R ( x ) = y 0 u + a 2 2 y 0 u ,
u = - D w 2 2 u ( 3 4 w - 1 / 4 - C ) f ,
u = w u - w u 2 u 2 ( 3 4 w - 1 / 4 - C ) - 3 32 w 2 w - 5 / 4 u ,
w = - D w 2 f ,
w = D w 3 ( 2 D f 2 - f w ) ,
f = 2 β [ exp ( - β 2 ) - exp ( - β 2 1 + γ ) ] ,
f = - f β - 4 [ exp ( - β 2 ) - 1 1 + γ · exp ( - β 2 1 + γ ) ] .
y t = - 1 ( n - 1 ) y ( x ) ( x t - x ) ,
x t = x - y ( x ) / y ( x ) ,
y t = - 1 ( n - 1 ) y ( x ) ,

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