Soliton-like molecular deformations in a nematic liquid crystal film

Physics Letters A 372 (2008) 2623–2633

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Soliton-like molecular deformations in a nematic liquid crystal film

M. Daniel a,∗, K. Gnanasekaran b

a Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024, Tamil Nadu, Indiab Department of Physics, Nehru Memorial College, Puthanampatti, Tiruchirapalli 621 007, Tamil Nadu, India

Received 9 August 2006; received in revised form 21 November 2007; accepted 27 November 2007

Available online 23 December 2007

Communicated by C.R. Doering

Abstract

We study the nature of molecular deformations in a nematic liquid crystal film with elastic energy under homeotropic boundary conditions. Thedeformation in terms of splay, twist and bend fields of the director axis is found to be governed by the completely integrable Davey–Stewartson-I(DS-I) equation in (2 + 1) dimensions. Using the line soliton and breather solutions of the DS-I equation, the director axis is constructed, thecomponents of which exhibit damped spatial oscillations. However, the splay and bend fields of the director axis exhibit localized structures ofdeformation.© 2008 Elsevier B.V. All rights reserved.

PACS: 61.30-v; 61.30.Gd; 02.30.Ik

1. Introduction

Liquid crystals are interesting anisotropic materials that have been investigated because of their importance in displays and alsothey serve as an interesting nonlinear model exhibiting different kinds of molecular deformations [1–5]. In particular, the theoreticalstudy of nematic liquid crystals play a very important role because of their simple geometric structure. In nematic liquid crystal, themolecules are considered as elongated rods, which are positionally disordered but ordered in orientation. The average direction oforientation of the liquid crystal molecules along a particular direction is represented by a vector field n(r) known as director axis[6,7]. However, director reorientation or molecular excitation in nematic liquid crystals takes place due to elastic deformations suchas splay, twist and bend [8]. In addition to the above hydrodynamic flow, externally applied electric, magnetic or optic field canalso induce director axis deformations [8]. In general, director axis deformation in nematic liquid crystals is nonlinear in nature andit was studied by Legar [9] by taking into account the twist energy deformation alone and under the influence of a static magneticfield. The nonlinear behaviour leads to soliton in nematic liquid crystal, which was studied by Lei et al. [10] under uniform shear.Further, soliton and pattern formation in nematic liquid crystals were obtained by Migler and Meyer [11] in the presence of arotating magnetic field. The above studies were carried out in one dimension and by assuming that the elastic coefficients due tosplay, twist and bend are equal (one constant approximation). The effect of nonlinearity on molecular deformations in nematicliquid crystal even in the absence of any externally applied field is also equally important. Recently, the present authors studiedthe director dynamics in a quasi-one-dimensional nematic liquid crystal under elastic deformations in the absence of any externallyapplied field without imposing the one constant approximation [12,13]. The molecular deformations in terms of a rotational directoraxis field (related to bend mode) in this case, is found to exhibit localized behaviour in the form of pulse, hole and shock as well assoliton and finally leading to damped oscillations of the director axis. The above studies were restricted to one spatial dimension ofthe material.

* Corresponding author.E-mail address: [email protected] (M. Daniel).

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2624 M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633

Further, in the case of nematic liquid crystal films (two spatial dimensions) director dynamics is represented in terms of planarsolitons in the presence of a constant magnetic field [14,15]. Also, two-dimensional behaviour of director axis in a nematic liquidcrystal has been studied under spherical approximation in the presence of a rotating magnetic field and stationary solitons wereobtained [16]. Also, molecular reorientation dynamics induced by circularly polarized, and spatially confined light beams leadsto the formation of various chaotic regimes such as periodic, quasi-periodic, and intermittent-types [17,18]. In the presence of aGaussian-type electric field, two-dimensional spatial soliton has been found within the inner region of the nematic film [19,20]. Inaddition, it has been proved experimentally that when a linearly polarized argon laser beam in the Gaussian mode is applied in adirection normal to the nematic sample of N − 4′-methoxybenzylinden and 4-cyano–4′ − N -hexylbiphenyl, strong self focusingaction occurs leading to the formation of soliton in planar geometry [21]. Further, it has been demonstrated experimentally thatnematic phase exhibits two-dimensional spatial solitary waves when acted upon by a few milliwatt power beam [3,22]. In thepresent Letter guided by an experience in the case of one-dimensional nematics with pure elastic energies, we try to generatesoliton-like molecular reorientations in the absence of any externally applied field in a nematic film. The plan of the Letter is asfollows. After deriving the dynamical equation by considering the elastic free energy in Section 2, we rewrite the same in terms ofa well-known completely integrable soliton equation in (2 + 1) dimensions, using a multi-scale expansion technique in Section 3.After presenting the soliton solutions of the nonlinear evolution equation in Section 4, we construct the director axis using Green’sfunction in Section 5. The results are concluded in Section 6.

2. Model and dynamical equation

We consider a nematic liquid crystal film of finite size (0l × 0m) with homeotropic boundary conditions (see Fig. 1). Thus onthe boundary walls, all the nematic liquid crystal molecules are arranged to point normal to the boundaries and are expected toremain invariant during the dynamical motion due to strong surface anchoring. However, the molecules in the inner region of thefilm are free to undergo spatial variation. The collective behaviour of the molecules may be described in terms of the dynamics ofthe director axis field n(r), with the constraint n · n = 1. The fluidity of the molecules gives rise to elastic deformations such assplay, twist and bend. The Frank free energy density associated with the above elastic deformations in the nematic film is givenby [8]

(1)f = 1

2

[K1(∇ · n)2 + K2(n · ∇ ∧ n)2 + K3

(n ∧ (∇ ∧ n)

)2],

where K1, K2 and K3 are the elastic constants associated with the splay, twist and bend deformations respectively and ∇ =e1

∂∂x

+ e2∂∂y

. The molecular field h corresponding to the above free energy can be obtained by minimizing the free energy density (1)

using hi = − ∂f∂ni

+ ∂j∂f

∂gji, i, j = x, y, z and gji = ∂jni . Thus, we obtain the expression for the molecular filed as

(2)h = K1∇(∇ . n) − K2[S(∇ ∧ n) + (∇ ∧ Sn)

] + K3[(

B ∧ (∇ ∧ n)) + (∇ ∧ (n ∧ B)

)],

where S = n .∇ ∧ n and B = n ∧ (∇ ∧ n). The thermodynamic force h acting on the director is given by h = h − (h · n)n. The term(h · n)n is the Lagrange multiplier term (projection of the molecular field on the director) which has been introduced in the fieldequation to take care of the condition n · n = 1 that explicitly states that in equilibrium the director must be at each point parallelto the molecular field. Any component of the field parallel to the director only serves the purpose of determining the Lagrangemultiplier but has no significance in the dynamics of the director which is the main objective of the study in this Letter [23].Further, the torque associated with this contribution of the field vanishes and hence we drop this term in our study because it is notof interest to us. In addition to the molecular field, there exists a viscous field due to pure rotational effect and another field arisingfrom the coupling of the director axis to the fluid motion given respectively by γ1(

∂n∂t

− ω ∧ n) and γ2An where γ1 and γ2 are the

viscosity coefficients. Here ω is the angular velocity of the background fluid and A is the velocity gradient tensor. However, the fluid

Fig. 1. A nematic liquid crystal film of finite size (0l × 0m) with homeotropic boundaries.

M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633 2625

velocity and its gradients are small enough such that the term proportional to γ2 can be neglected and hence the director motionis largely unaffected by the hydrodynamic flow. Also, the effect of background flow is negligible. Under these circumstances thedirector dynamics is described by the equation

(3)γ1∂n∂t

= K1∇(∇ . n) − K2[S(∇ ∧ n) + (∇ ∧ Sn)

] + K3[(

B ∧ (∇ ∧ n)) + (∇ ∧ (n ∧ B)

)].

The director dynamics is thus governed by a highly nontrivial vector nonlinear partial differential equation and solving the same inits natural vector form is a challenging task.

Hence, we rewrite Eq. (3) by taking curl and divergence on both sides separately and by introducing two new fields namely Aand Φ such that A ≡ (Ax,Ay,Az) = (∇ ∧ n) and Φ = ∇ · n which are related to the bend and splay modes respectively

(4a)γ1∂A∂t

(r, t) = K3(∇2A − A2A

),

(4b)γ1∂Φ

∂t(r, t) = K1∇2Φ − K3A2Φ.

While deriving Eqs. (4), it is assumed that the rotation of the director axis (A) is always normal to n so that n · A = 0 which makesthe term proportional to K2 to vanish. In order to associate Eqs. (4) with more standard evolution equations, we introduce a newcomplex scalar field in terms of the components of A. For this, we make A to be a two component vector in the xy-plane, i.e.,A = (Ax,Ay,0) and define a new complex scalar field as U = Ax + iAy . When Az ≡ (∇ ∧ n)z = 0, we have ∂nx

∂y= ∂ny

∂xand the

constraint n · A = 0 leads to Ay = −nx

ny Ax . Also in this process, the number of unknowns that appear in the resultant equations(Eqs. (5a), (5b)) are reduced by one. Using the above, Eqs. (4) after suitable rescaling of time can be written as

(5a)∂U

∂t= (∇2U − |U |2U)

,

(5b)∂Φ

∂t= K1

K3∇2Φ − |U |2Φ.

The above set of coupled equations (5) describes the director dynamics in terms of two new fields namely U and Φ which arerespectively related to the bend and splay modes. The advantage of rewriting the dynamical equation in terms of these new fieldslies in associating these set of new equations with a family of known integrable nonlinear evolution equations.

3. Molecular deformation dynamics

Having derived the dynamical equation in a convenient form, the problem now boils down to solving Eqs. (5) to understandthe director dynamics. As we have considered a nematic liquid crystal film for our study, we fix the film to lie in the xy-plane

so that ∇2 ≡ ∇⊥2 = ∂2

∂x2 + ∂2

∂y2 . We assume the solutions of Eqs. (5) in the form U = ∑∞p=0 εαpqp(x, y, t)e−ε2pψ(x,y)t and Φ =∑∞

p=0 εαpvp(t) e−ε2pψ(x,y)t , where ε is an arbitrary small parameter and αp’s are real integers. We substitute the above expansions

for U and Φ in Eqs. (5), rescale the time and spatial variables as t → −ε2t , x → √2εx and y → √

2εy, and equate the coefficientsof every eε2pψ(x,y)t separately for different p values to zero. Thus, the coefficients of the terms proportional to eε2nψ(x,y)t where n

is an integer give

ε2+αn

[∂qn

∂t+ nψqn + 1

2∇⊥2qn − 1

2ε2n

{(∇⊥2ψ)qn +

(∂qn

∂x

∂ψ

∂x+ ∂qn

∂y

∂ψ

∂y

)t + 2ε2nt

((∂ψ

∂x

)2

+(

∂ψ

∂y

)2)qn

}]

(6a)−∑

s

∑r

εαs εαn−r εαr−s qn−rqr−sq∗s = 0,

and

ε2+αn

[∂vn

∂t+ nψvn + K1

2K3ε2nvnt

{∇⊥2ψ + 2ε2nt

{(∂ψ

∂x

)2

+(

∂ψ

∂y

)2}}]

(6b)−∞∑

s=0

∞∑r=0

εαs εαn−r εαr−s qn−rq∗r−svs = 0.

Operating Eq. (6b) by the operator (εαn ∂2

∂y2 − ∂2

∂x2 ) we obtain

ε2(1+αn) ∂2

2− ε2+αn

∂2

2

[∂vn + nψvn + K1

ε2nvnt

{∇⊥2ψ + 2ε2nt

{(∂ψ

)2

+(

∂ψ)2}}]

∂y ∂x ∂t 2K3 ∂x ∂y


(6c)−∞∑

s=0

∞∑r=0

εαs εαn−r εαr−s

[∂2

∂x2− εαs εαn−r εαr−s

∂2

∂y2

]qn−rq

∗r−svs = 0.

Eqs. (6) are in the form of an infinite series in powers of ε and we assume that for every value of n, ε → 0. However, the termscorresponding to the minimal power of the parameter ε vanishes and consequently we have [24]

(7a)i∂q1

∂t+ 1

2∇⊥2q1 + |q1|2q1 + ψq1 = 0,

(7b)∂2ψ

∂y2− ∂2ψ

∂x2− 2

∂2|q1|2∂x2

= 0.

While writing the above equations we have transformed time variable as t → it . Also we have chosen qr−s = −qs−r , qs = q−s ,vs = v−s and further we have assumed αn = α−n = n for n � 1 and α0 = 2.

The set of coupled equations (7) are together known as Davey–Stewartson-I (DS-I) equation, which was first derived in fluiddynamics describing the evolution of waves of slowly varying amplitude on a two-dimensional water surface under gravity [25].The above equation later appeared in several other contexts as well including stability of gravity waves [26], nonlinear optics [27],plasma [28], spin systems [29], etc.

4. Soliton in a nematic film

The DS-I equation given in Eqs. (7) is a two-dimensional integrable generalization of the completely integrable cubic nonlinearSchrödinger (NLS) equation and admits solutions in the form of line soliton, dromion and breather which were originally obtainedby solving the equation using dressing method in nonlocal Riemann–Hilbert problems [30]. It is also possible to construct theN -soliton solutions for the equation using Hirota’s bilinearization procedure [29,31,32]. However, as our problem involves onlytime-independent boundary conditions, the dromion solution (which involves time-dependent boundary conditions) is not relevantto us here. Therefore in the following, we write down the explicit form of line one soliton and (1,1) breather solutions obtainedthrough Hirota’s bilinear method.

4.1. Line one soliton

The line one soliton solutions of the DS-I equation (7) is written as

(8a)q1 = 4√

λRμR sech(χR + ln(4

√λRμR )

)exp(iχI ),

where

(8b)χR = (λR + μR)x + (λR − μR)y − 2[λRλI + μRμI ]t + χ(0)R ,

(8c)χI = (λI + μI )x + (λI − μI )y + [(λR

2 + μR2) − (

λI2 + μI

2)]t + χ(0)I .

Here χR and χI are the real and imaginary parts of χ = (λ + μ)x + (λ − μ)y + i[λ2 + μ2]t + χ(0) and λR(λI ), μR(μI ) and χ(0)R

(χ(0)I ) are the real (imaginary) part of the complex constants λ, μ and χ respectively. Using the line one soliton solution given in

Eq. (8a) in Eq. (7b) the field ψ(x, y) can be obtained by solving the resultant equation. The result reads as

(9)ψl = −16λRμR

[2

λR + μR

tanh((λR + μR)x + (λR − μR)y + δ

) − sech2((λR + μR)x + (λR − μR)y + δ)]

,

where δ = ln(4√

λRμR ). Knowing q1 and ψ , the bend field U and hence A can be constructed using the relation U ≡ Ax + iAy =q1e

−ψt . Similarly, the splay field Φ can be obtained using the relation Φ = v1e−ψt . However, since we have switched to an

imaginary time variable (t → it) while going from Eq. (6) to Eq. (7), this will arise problem in the case of the solution with realtime. This can be avoided by choosing |λR| = |λI | and |μR| = |μI |. Therefore, Eq. (8c) must be replaced by χI = (λI + μI )x +(λI − μI )y + χ

(0)I . Thus, using Eqs. (8) and (9) we obtain the bend and splay fields as

U ≡ Ax + iAy

= 4√

λRμR sech(χR + ln(4

√λRμR )

)[exp

[16λRμR

(2

λR + μR


)

(10a)− 16 sech2((λR + μR)x + (λR − μR)y + δ))

t

]]exp iχI ,


and

(10b)

Φ = v1 exp

[16λRμR

(2

λR + μR


) − sech2((λR + μR)x + (λR − μR)y + δ))

t

].

Eq. (10a) represents the bend field (i.e., rotation of the director axis) and (10b) the splay field (i.e., divergence of the director axis)of the nematic film corresponding to the line soliton solution of DS-I equation. In Fig. 2(a), we have given a snapshot (at t = 1)of the line one soliton of the DS-I equation as found in Eq. (8) by choosing λR = 0.012, μR = 1.0, λI = 0.012 and μI = 1.0. InFig. 2(b), we have plotted |A|2 (Eq. (10a)) which is related to the stationary bend mode corresponding to the line one soliton for thesame parameter values used to plot the line one soliton in Fig. 2(a). In Fig. 2(c), Φ , which is related to the stationary splay modecorresponding to the line soliton (Eq. (10b)) is plotted by choosing λR = 0.012, μR = 1, λI = 0.012, μI = 1 and v1 = 0.001. FromFigs. 2(b), (c) we observe that the bend field |A|2 which represents the rotation of the director axis and the splay field that is relatedto the divergence of the director axis corresponding to the line one soliton exhibit localized behaviour in the form of pulse and kinkrespectively.

4.2. (1,1) breather solution

The (1,1) breather solution of the DS-I equation (7a) and (7b) which is localized spatially and oscillatory in time is given by

(11)q1 = ρe(λ′R+μR

′)x+(λ′R−μR

′)y

1 + je2λ′R(x+y) + ke2λ′

R(x−y) + le(2λ′R(x+y)+2λ′

R(x−y))ei(λ′

R2+μ′

R2)t ,

where ρ is a complex constant and j , k, l and λ′R , μ′

R , are real positive constants such that λ′R

2 + μ′R

2 � 1. Knowing q1, the fieldψ(x, y) can be constructed by substituting the solution q1 in Eq. (7b) and the resultant equation solved. Thus, the field ψb(x, y)

corresponding to the (1,1) breather solution given in Eq. (11) is written as

(12a)ψb = −ρ2e2λ′Rx

[N

D− 4

{1 + eλ′

Rx(coshλ′

Ry + eλ′Rx

)}−2],

where N = tanh(λ′Ry) + 2{(eλ′

Rx − 2) + (1 + 12 − e

5λ′R

2 x sinhλ′

R

2 x − 4 coshλ′Rx) sech2 λ′

Ry} arctan[ tanhλ′Ry

2(1+e32 λ′

Rx cosh

λ′R2 x)

] and D =2{( 19

2 − 4coshλ′Rx − 4e

9λ′R

2 x sinhλ′

R

2 x − 2e5λ′

R2 x cosh

λ′R

2 x − 5e2λ′Rx) sech2 λ′

Ry − (e4λ′Rx + 3eλ′

Rx − 4)}. Having found q1 and ψ , thefield U and hence A can be constructed using the relation U ≡ Ax + iAy = q1e

−ψt . Similarly, the field Φ can be obtained usingthe relation Φ = v1e

−ψ(x,y)t . Thus, in the case of (1,1) breather solution (Eqs. (11) and (12a)), we obtain

U = ρe(λ′R+λ′

R)x+(λ′R−λ′

R)y

1 + je2λ′R(x+y) + ke2μ′

R(x−y) + le2(λ′R+λ′

R)x+2(λ′R−λ′

R)y

(13a)× exp

[ρ2e2λ′

Rx

[N

D− 4

{1 + eλ′

Rx(coshλ′

Ry + eλ′Rx

)}−2]

+ i(λ′

R2 + λ′

R2)

t

],

and

(13b)Φ = v1 exp

[ρ2 e2λ′

Rx

[N

D− 4

{1 + eλ′

Rx(coshλ′

Ry + eλ′Rx

)}−2]t

].

In Fig. 2(d), we have plotted a snapshot (at t = 1) of the (1,1) breather solution of the DS-I equation as found in Eq. (11)by choosing λ′

R = 0.012, μ′R = 0.01 and ρ = 0.001. In Fig. 2(e), |A|2 (Eq. (13a)) which is related to the stationary bend mode

corresponding to (1,1) breather solution of the DS-I equation has been plotted for the same parameter values. In Fig. 2(f), Φ ,which is related to the stationary splay mode (Eq. (13b)) corresponding to (1,1) breather solution of DS-I equation is plotted bychoosing λ′

R = 0.012, μ′R = 1.0 and v1 = 0.001. The (1,1) breather solution of the DS-I equation is known to exhibit a highly

localized pattern (see Fig. 2(d)). From Fig. 2(e) and 2(f) we observe that the bend field |A|2 (rotation of the director axis) is stillexhibiting a localized behaviour similar to breathers with an added effect of a localized spike at a specified location (which isevident from Fig. 2(f)), that represents the splay field Φ (divergence of the director axis) corresponding to (1,1) breather solutionof the DS-I equation.

5. Construction of director axis

Having found the fields A = ∇ ∧ n and Φ = ∇ · n which correspond to the bend and splay modes respectively in terms of theline one soliton and (1,1) breather solutions of the DS-I equation we now construct the director axis n using these results. The


(a) (b)

(c) (d)

(e) (f)

Fig. 2. A snapshot of (a) line one soliton |q1|2 (Eq. (8)) of DS-I equation for λR = 0.012, μR = 1, λI = 0.012 and μI = 1, (b) of the bend field (|A|2) (Eq. (10a))corresponds to the line one soliton for the same parameter values as given above and (c) splay mode (Φ) (Eq. (10b)) corresponding to the line one soliton of DS-Iequation for λR = 0.012, μR = 1, λI = 0.012, μI = 1 and v1 = 0.001, (d) (1,1) breather solution of DS-I equation |q1|2 (Eq. (11)), for λ′

R= 0.012, μ′

R= 0.01

and ρ = 0.001. (e) the bend mode (|A|2) (Eq. (13a)) corresponding to (1,1) breather solutions of DS-I equation for the same parameter values as in (d) and (f) splaymode (Φ) (Eq. (13b)) corresponding to (1,1) breather solution of DS-I equation for λ′

R= 0.012, μ′

R= 0.01 and v1 = 0.001.

equations defining A and Φ can be combined and written as the following two-dimensional Poisson equation:

(14)∇⊥2n = ∇Φ − ∇ ∧ A, n = (nx,ny, nz

).

The homeotropic boundary conditions for our problem is written as

(15)nx = 1, ny = 1, nz = 0|x=(0,l);y=(0,m).

The above inhomogeneous boundary conditions (associated with the x- and y-components of the director axis) can be convertedinto homogeneous ones by defining a new vector field p ≡ (px,py,pz) such that px = nx − 1, py = ny − 1, and pz = nz. In termsof the new vector field, the two-dimensional Poisson equation is written in the component form as

(16)∇⊥2px = ∂Φ

∂x, ∇⊥2py = ∂Φ

∂y, ∇⊥2pz = ∂Ax

∂y− ∂Ay

∂x.

The above Poisson equations with the set of new homogeneous boundary conditions can be solved using the Green’s functionmethod of eigenfunction expansion.


The equation governing the Green’s function G is written as

(17)∇⊥2G = −δ(x − ζ )δ(y − β),

with G satisfying the boundary conditions G(x, ζ ;y,β) = 0|(x=0,l);(y=0,m). Here ζ and β are arbitrary points along x and y

directions respectively. The Green’s function solution of Eq. (17) can be expressed using the method of eigenfunction expansionas [33]

(18)G(x, ζ ;y,β) = 4lm

π2

∞∑s=1

∞∑s′=1

sin sπl

ζ sin s′πm

β sin sπl

x sin s′πm

y

(ms′)2 + (ls)2.

Using the value of the above Green’s function G, the components of the vector field p namely px , py and pz can be constructedusing the right-hand sides of Eq. (16) as follows:

(19)pi(x, y) =x∫

−∞

y∫−∞

dx′ dy′G(x′, y′; ζ,β)Ri , i = x, y, z,

where Ri ’s are the right-hand sides of the px , py and pz equations in Eqs. (16). On substituting the Green’s function G fromEq. (18) and the values of Φ , Ax and Ay from Eqs. (10) corresponding to the line one soliton of the DS-I equation in Eq. (19), weobtain the values of px , py and pz as follows:

(20a)px = γss′[hx(x, y, t) − Hx(x, y, t)

],

(20b)py = γss′[hy(x, y, t) − Hy(x, y, t)

],

(20c)pz = γss′[hz(x, y, t) + h′

z(x, y, t) − Hz(x, y, t) − H ′z(x, y, t)

].

In the above equations γss′ = 4lm

π2

∑∞s=1

∑∞s′=1 sin sπ

lζ sin s′π

mβ . Defining

f (x, y, t) = sinsπ

lx sin

s′πm

y exp

[16λRμR

(2

λR + μR

(tanh

((λR + μR)x + (λR − μR)y

) + δ)

− sech2((λR + μR)x′ + (λR − μR)y′ + δ))

t

]

the functions hi(x, y, t) and Hi(x, y, t) (i = x, y, z) are written as

(21a)hx(x, y, t) = v1

y∫−∞

f (x, y′, t) dy′, hy(x, y, t) = v1

x∫−∞

f (x′, y, t) dx′,

(21b)Hx(x, y, t) = v1

x∫−∞

y∫−∞

f (x′, y′, t) sinn′πm

y′ dx′ dy′,

(21c)Hy(x, y, t) = v1

x∫−∞

y∫−∞

f (x′, y′, t) sinnπ

lx′ dx′ dy′,

(21d)hz(x, y, t) =x∫

−∞f (x′, y, t)

[sech2((λR + μR)x′ + (λR − μR)y + δ

)]cos

((λR + μR)x′ + (λR − μR)y

)dx′,

(21e)h′z(x, y, t) =

y∫−∞

f (x, y′, t)[sech2((λR + μR)x′ + (λR − μR)y′ + δ

)]sin

((λR + μR)x + (λR − μR)y′)dy′,

(21f)

Hz(x, y, t) =x∫

−∞

y∫−∞

f (x′, y′, t) sinnπ

lx′ cos

(n′πm

y′)

cos((λR + μR)x′ + (λR − μR)y′)

× [sech2((λR + μR)x′ + (λR − μR)y′ + δ

)]dx′ dy′,

(21g)

H ′z(x, y, t) =

x∫−∞

y∫−∞

f (x′, y′, t) cosnπ

lx′ sin

n′πm

y′ sin((λR + μR)x′ + (λR − μR)y′)

× [sech2((λR + μR)x′ + (λR − μR)y′ + δ

)]dx′ dy′.


As it is difficult to evaluate the above integrals in the present form, we expand the exponential function that appear in the functionf (x, y, t). After using the expansion and on evaluating the integrals at the lowest order (dropping higher order terms) and using theresults in Eqs. (20), we obtain the components of the director axis in the form

(22a)

nx ≡ px + 1

= 1 + γss′v1m

s′π

(l

sπ

)(1 − 16λRμR)

[sin

sπ

lx cos

s′πm

y

+ exp

[16λRμR

(2

λR + μR

tanh((λR + μR)x + δ

) − sech2((λR + μR)x + δ))

t

]],

(22b)

ny ≡ py + 1 = 1 + γss′v1m

sπ(1 − 16λRμR)

(1 − m

s′π

)[cos

sπ

lx sin

s′πm

y

+ exp

[16λRμR

(2

λR + μR


) − sech2((λR + μR)x + δ))

t

]],

and

nz ≡ pz = γss′ l

sπ(16λRμR − 1)

(1 − m

s′π

)cos

sπ

lx sin

s′πm

y

+ m

s′πsin

((λR + μR)x

)sin

sπ

lx cos

s′πm

y sech((λR + μR)x + δ

)

(22c)× exp

[16λRμR

(2

λR + μR


) − sech2((λR + μR)x + δ))

t

].

The above solutions must satisfy the constraint n · A ≡ n · ∇ ∧ n = 0, which is equivalent to nxAx + nyAy = 0. Therefore, toverify this, we evaluate the quantities nxAx and nyAy separately using Eqs. (22a), (22b) and (10a) and write the final form of theexpression nxAx + nyAy as

nxAx + nyAy

= 16lm

π2

√λRμR(1 − 16λRμR)

∞∑s=1

∞∑s′=1

l

π ssin

πζs

lsin

πβs′

m

×{[

exp

[16λRμR

(2

λR + μR


) − sech2((λR + μR)x + δ))

t

]+ sin

π s

lx cos

π s′

my

]cos χ1

+ π s

l

(1 − m

π s′

)[exp

[16λRμR

(2

λR + μR


) − sech2((λR + μR)x + δ))

t

]

+ cosπs

lx sin

πs′

my

]sinχ1

}

× sech(χR + ln(4

√λRμR )

)exp

[16λRμR

(2

λR + μR


)

(23)− sech2((λR + μR)x + (λR − μR)y + δ))

t

],

which should vanish. The right-hand side of Eq. (23) vanishes when∑∞

s=1∑∞

s′=1l

π ssin πζs

lsin πβs′

m= 0 which is equivalent to∑∞

s=1∑∞

s′=1l

πs[cos(πζs

l+ πβs′

m)− cos(πζs

l− πβs′

m)] = 0. The left-hand side of the above condition averages out to be zero quickly

when l and m are large, that is in the case of an infinite film. Thus, the solutions satisfy the constraint n · A = 0 in the limit of aninfinite film.

We repeat the entire calculations for the case of (1,1) breather solution of the DS-I equation. However, in this case, the variousresultant expressions are very complicated and due to the unwieldy nature of the final results, we are not presenting the detailshere. In Figs. 3(a)–(c), we have given a snapshot of the x-, y- and z-components of the director axis namely nx , ny and nz fors = s′ = 1.0, λR = 0.012, μR = 1.0, λI = 0.012, μI = 1.0, ζ = β = 0.1, and l = m = 1.0 at time t = 1. From the figures itis observed that the x- and y-components of the director axis namely nx and ny slowly gain spatial oscillations in the xy-planestarting from the equilibrium configuration. On the other hand, the z-component of the director axis (nz) exhibits uniform periodicoscillations along the y-direction. However, along the x-direction the amplitude grows and gets saturated. Further, when we analyzethe long time behaviour of the director axis oscillations along different directions, we find that the oscillations are getting dampedas shown in Figs. 3(d)–(i).


(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i)

Fig. 3. Snapshots of the x-, y- and z-components (nx,ny,nz) of the director axis (Eqs. (22a)–(22c) respectively), exhibiting spatial oscillations for s = s′ = 1.0,λR = 0.012, μR = 1.0, λI = 0.012, μI = 1.0, v1 = 0.001, ζ = β = 0.1 and l = m = 1.0 at t = 1. (d)–(i): Long-time behaviour of the x-, y- and z-components (nx ,ny and nz) of the director axis along x and y axes showing damped oscillations for the same values of the parameters.


6. Conclusions

In this Letter, we have investigated the nature of director dynamics in a nematic liquid crystal film, with homeotropic boundaryconditions. Starting from the Frank free energy density for splay, twist and bend-type elastic deformations, we constructed theassociated field equation by minimizing the free energy density. In order to solve the resultant field equation for understanding thedirector dynamics, we introduced two new fields in terms of splay and bend modes (i.e., divergence and rotation of the directoraxis respectively). A multi-scale expansion was found useful in finally writing the resultant equation in terms of the well-knowncompletely integrable Davey–Stewartson equation in (2 + 1) dimensions which admits line soliton and breather solutions for ourproblem. Using these soliton solutions of DS-I equation, the components of the director axis were constructed by solving thePoisson equation associated with the definitions of splay and bend fields of our model by using Green’s function technique.

From the results, we observe that the splay and bend fields of the director axis exhibit localized structures of reorientation dueto elastic deformations in close analogy with the localized soliton and breather solutions of the DS-I equation. Spatial soliton innematic liquid crystal was experimentally observed when the nematic film is allowed to pass through high intense coherent laserbeam [3,21,34]. However, there is no experimental evidence available in the literature for the generation of solitons induced throughelastic deformations. This may be due to the fact that solitons generated through elastic deformations are expected to be less stablecompared to solitons generated through externally applied optic field. Hence, it may be very difficult to observe soliton inducedthrough elastic deformation. It is planned to carry out stability analysis of the elastic solitons and the results will be publishedelsewhere.

From Fig. 3, we notice that at a given time the x- and y-components of the director axis gain uniform oscillations form theirequilibrium configuration. However, the z-component of the director axis shows a slightly different behaviour with the amplitudegrowing steadily and gets saturated along x-direction but exhibiting uniform oscillations along y-direction. However, the longtime behaviour of oscillation of the director axis components in different directions is shown to exhibit damping feature of theoscillations, which is due to the fact that the two-dimensional Poisson equation is solved using the eigen function expansion methodfor Green’s function.

Acknowledgements

The work of M.D. forms part of a major DST project. K.G. would like to thank UGC for a Teacher Fellowship under the FIPprogramme.

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Soliton-like molecular deformations in a nematic liquid crystal film