THE SOLITON SOLUTIONS OF SOME

NONLINEAR DYNAMICAL MODEL

EQUATIONS

SUMMARY

OF THE

THESIS

SUBMITTED TO

THE FACULTY OF SCIENCE

KURUKSHETRA UNIVERSITY, KURUKSHETRA

FOR THE DEGREE

OF

DOCTOR OF PHILOSOPHY IN

PHYSICS

BY

HITENDER KUMAR

DEPARTMENT OF PHYSICS

KURUKSHETRA UNIVERSITY

KURUKSHETRA ‒ 136 119, INDIA

July ‒ 2013

1 Introduction

The origin of nonlinear partial differential equations (PDEs) is very old. They had undergone ma-

jor new developments during the second half of the twentieth century. One of the main needs for

developing nonlinear PDEs has been the study of nonlinear wave propagation problems. These

problems arise in diverse areas of applied mathematics, physics and engineering, including fluid

dynamics, nonlinear optics, solid mechanics, plasma physics, quantum field theory and condensed-

matter physics. Nonlinear wave equations, in particular, have provided several examples of new

solutions that are remarkably different from those obtained for linear wave problems. The best

known examples of these are the corresponding shock waves, water waves, solitons and solitary

waves. Indeed, the theory of nonlinear waves and soliions has experienced a revolution over the

past few decades. During this revolution, many remarkable and unexpected phenomena have also

been observed in physical, chemical, and biological systems. Other major achievements include

the discovery of soliton interactions, the inverse scattering transform method for finding the ex-

plicit exact solution for several PDEs and asymptotic perturbation analysis for the investigation of

nonlinear evolution equations.

Historically, the famous 1965 paper of Zabusky and Kruskal marked the birth of the new concept

of the soliton, a name intended to signify particle like quantities [1]. Subsequently, Zabusky (1967)

[2] confirmed, numerically, the actual physical interaction of two solitons, and Lax (1968) [3] gave

a rigorous analytical proof that the identities of two distinct solitons are preserved through the

nonlinear interaction governed by the KdV equation. Physically, when two solitons of different

amplitudes (and hence, of different speeds) are placed far apart on the real line, the taller (faster)

wave to the left of the shorter (slower), the taller one eventually catches up to the shorter one and,

then, overtakes it. When this happens, they undergo a nonlinear interaction according to the KdV

equation and emerge from the interaction completely preserved in form and speed with only a phase

shift. These discoveries have, in turn, led to extensive theoretical, experimental, and computational

studies over the last few years. Many nonlinear model equations have now been found in nonlinear

science to explain many of the novel features of dynamical systems.

Keeping in view many interesting applications of nonlinear PDEs in nonlinear dynamics, the sub-

ject matter of present thesis is to investigate analytic soliton solutions of some nonlinear PDEs

having vital applications in different fields. So to achieve this goal, newly developed analytic tech-

niques have been utilized and a variety of soliton solutions are obtained.

1

2 Methodology

Nonlinear evolution equations are widely used as models to describe complex physical phenomena

in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics and

chemical physics. Given a nonlinear PDE, there is no general way of knowing whether it has soliton

solutions or not, or how the soliton solutions can be found. In order to get a better understanding

of the underlying phenomena as well as their further applications in practical life, it is important

to seek their exact solutions. Analytical solutions to nonlinear PDEs play an important role in

nonlinear science, especially in nonlinear physical science since they can provide much physical

information and more insight into the physical aspects of the problem and thus lead to further ap-

plications. Moreover, new exact solutions may help researchers to find new phenomena. The exact

solutions, if available, of nonlinear PDEs facilitate the verification of results for numerical solvers

and aid in the stability analysis of solutions.

A literature survey reveals that, researchers usually employed a variety of distinct methods to ana-

lyze nonlinear evolution equations. These methods range from reasonable to difficult that require

a huge size of work. In fact there is no unified method that can be used for all types of nonlinear

evolution equations. For single soliton solutions, several methods, such as the inverse scatter-

ing transformation method [4], Hirota’s bilinear method [5], the truncated Painleve expansion [6],

Backlund transformation method [6], Darboux transformation method [7] homogeneous balance

method [8], projective Riccati equation method [9], Jacobi elliptic functions method [10] and many

others have been used in past. Lot of informations and details about these methods are presented

in several texts. Therefore, in order to solve some practical problems, one should select an ap-

propriate method from the list of available methods. Here, we describe main features of some of

the commonly used direct methods for solving nonlinear PDEs which are also used in the present

work.

2.1 The tanh-coth method

The main features of this method can be summarized as follows. To find the traveling wave so-

lutions, a nonlinear PDE of the form P (u, ut, ux, utt, uxt, uxx......) = 0, can be converted into an

ordinary differential equation (ODE) by introducing the wave variable ξ = (x − ct). Now, if one

introduces a new independent variable Y = tanh(µξ) where µ is the wave number, leads to the

2

change of derivatives:

d

dξ= µ(1− Y 2)

d

dY,

d2

dξ2= −2µ2Y (1− Y 2)

d

dY+ µ2(1− Y 2)

d2

dY 2,

d3

dξ3= 2µ3Y (1− Y 2)(3Y 2 − 1)

d

dY− 6Y µ3(1− Y 2)2

d2

dY 2

+ µ3(1− Y 2)3d3

dY 3, (1)

and so on. The tanh-coth method admits the use of the finite expansion

u(ξ) = S(Y ) =M∑n=0

anYn +

M∑n=1

bnY−n. (2)

where M is a positive integer to be determined. To determine the parameter M , balance the linear

terms of highest order in the resulting equation with the highest order nonlinear terms. For nonin-

teger M a transformation formula is used to overcome this difficulty. Then collect all coefficients

of powers of Y in the resulting equation where these coefficients have to vanish. This will give a

system of algebraic equations involving the parameters an, bn, µ, ξ and c. Having determined these

parameters, an analytic solution u(x, t) in a closed form is obtained. Note that the expansion (2)

reduces to the standard tanh method for bn = 0, 1 ≤ k ≤ M . The solutions obtain by this method

may be solitons in terms of sech2, or may be kinks in terms of tanh. This method may also give

periodic solutions as well. This method is used in chapter 4 to solve nonlinear diffusion problems.

In past this method has been successfully applied to a large number of nonlinear PDEs [11].

2.2 F-expansion method

For a given nonlinear evolution equation, say, for two variables x and t

P (u, ut, ux, utt, uxt, uxx, ...) = 0. (3)

We assume that Eq.(3) has the following formal solutions

u(x, t) = a0 +l∑

i=1

[aiF (ξ)i + biF (ξ)−i], (4)

where a0 = a0(x, t), ai = ai(x, t), bi = bi(x, t), (i = 1, ..., l), ξ = ξ(x, t) are all arbitrary functions

of indicated variables and F (ξ) satisfies the following elliptic equation [10](dFdξ

)2= c0 + c2F

2 + c4F4, (5)

3

where c0, c2 and c4 are real constants related to the square of elliptic modulus m of Jacobi elliptic

functions. When modulus number m → 0 or 1(0 < m < 1), we can get trigonometric function

solutions or hyperbolic function solutions respectively.

Determine the parameter l by balancing the highest order derivative terms with the nonlinear terms

in Eq.(3). It represents a subtle balance of the dissipation effect and the dispersion effect in physics

where soliton origins from.

Substituting Eqs.(4) and (5) into Eq.(3) yields a set of algebraic polynomials for F (ξ). Eliminating

all the coefficients of the powers of F (ξ) and√c0 + c2F 2 + c4F 4 yields a series of differential

equations, from which the parameters a0, ai, bi(i = 1, ..., l) and ξ are explicitly determined.

Substituting a0, ai, bi and ξ into Eq.(4) and selecting the Jacobian elliptic functions, we can derive

all kinds of Jacobian elliptic function solutions of Eq.(3). As a matter of fact, this method has been

applied to a number of specific cases involving both coupled and uncoupled nonlinear PDEs as well

as to variable coefficients nonlinear PDEs [10, 12].

The extended F-expansion method:

We now describe the extended F-expansion method for nonlinear evolution equations. We concisely

show what is extended F-expansion method and how to use it to find various periodic wave solutions

to nonlinear wave equations.

In this method, a nonlinear PDE

P (u, ut, ux, uy, uxt, utt, uyt, uxx, ...) = 0, (6)

can be converted to nonlinear ODE

O(u, u′, u′′, u′′′, ...) = 0, (7)

upon using a wave variable ξ = λ1x1+λ2x2+λ3x3+ ....λlxl−ωt, so that u(x1, x2, x3...., t) = u(ξ)

and the localized wave solution u(ξ) travels with speed of ω.

Now suppose that u(ξ) can be expanded as follows

u(ξ) =n∑

j=0

j∑i=0

cjiFi(ξ)Gj−i(ξ), cnn = 0, (8)

or

u(ξ) = a0 +n∑

i=1

aiFi(ξ) +

n∑j=1

bjF−j(ξ), an = 0, (9)

where cji, a0, ai and bj are constants to be determined and F (ξ) and G(ξ) satisfy the following

relations

F′2(ξ) = P1F

4(ξ) +Q1F2(ξ) +R1, G

′2(ξ) = P2G

4(ξ) +Q2G2(ξ) +R2, (10)

G2(ξ) = µF 2(ξ) + ν,R1 =Q2

1 −Q22 + 3P2R2

3P1

, µ =P1

P2

, ν =Q1 −Q2

3P2

, ν = 0.

(11)

4

The integer n is determined by considering the homogeneous balance between the governing non-

linear terms and highest order partial derivatives of u in Eq.(7).

Substituting Eq(8) or Eq.(9) into Eq.(7) and using Eqs.(10) and (11), we obtain a series in

F pGq(p = 0, 1, 2, ...l; q = 0, 1) or (F p, p = 0, 1, 2, ...l). Equating each coefficient of F pGq or

(F p) to zero yields a system of algebraic equations for cji (j = 0, 1, 2, ...n; i = 0, ..., j) and λi, ω,

or (ai, bj, i = 1, 2, ...n; j = 1, 2, ...n;λi, ω).

Now solving these equations by use of either Mathematica or Maple, cij , λi and ω can be expressed

in terms of Pi, Qi, Ri, µ, ν and the parameters of Eq.(7). Substituting these results into Eq.(8) or

(9) gives the general form of traveling wave solutions.

By using the relations (10) and (11), the appropriate kinds of the Jacobi elliptic function solutions

of Eq.(6) including the single functions and the combined function solutions are found. As we

know, when m → 1, Jacobi elliptic functions degenerate as hyperbolic functions and m → 0,

Jacobi elliptic functions degenerate as trigonometric functions. So we can get the corresponding

hyperbolic function solutions and trigonometric function solutions. We employ this method to the

nonlinear problems in third and fourth chapters.

2.3 The projective Riccati equation method

The well known projective Riccati equations read as

f′(ξ) = pf(ξ)g(ξ), (12)

g′(ξ) = R + pg2(ξ)− rf(ξ), (13)

where p is a nonzero real constant, R and r are two real constants. The relation between f and g is

g2 = −p[R− 2rf +

r2 + δ

Rf 2], δ = ±1, R = 0. (14)

In this method, a given nonlinear PDE in the unknown u(x, y, z, ..., t), is reduced into a ODE by

the traveling wave reduction u(x, y, z, ..., t) −→ u(ξ = λ1x+ λ2y+ λ3z+ ...+ λnt). The solution

u(ξ) is assumed of the form

u(ξ) =n∑

i=1

f i−1(ξ)[Aif(ξ) +Big(ξ)] + A0, R = 0, (15)

where Ai and Bi are constants to be fixed later and f(ξ) and g(ξ) are solutions of Eqs.(12) and (13).

The parameter n in Eq.(15) can be determined by balancing procedure. Now substituting Eq.(15)

with along conditions (12), (13) and (14) into O(u, u′, u

′′, ...) = 0, and setting the coefficient of

f igj(j = 0, 1, i = 0, 1, 2, 3, ...) to zero yields a set of determined over algebraic equations, from

5

which the constants Ai, Bi, R, r and λi can be found explicitly.

The exact solutions of Eqs.(12) and (13) are of the form [13, 14]

Case 1: When p = −1, δ = −1, R = 0

f1(ξ) =R sech(

√Rξ)

r sech(√Rξ) + 1

, g1(ξ) =

√R tanh(

√Rξ)

r sech(√Rξ) + 1

. (16)

Case 2: When p = −1, δ = 1, R = 0

f2(ξ) =R csch(

√Rξ)

r csch(√Rξ) + 1

, g2(ξ) =

√R coth(

√Rξ)

r csch(√Rξ) + 1

. (17)

Case 3: When p = 1, δ = −1, R = 0

f3(ξ) =R sec(

√Rξ)

r sec(√Rξ) + 1

, g3(ξ) =

√R tan(

√Rξ)

r sec(√Rξ) + 1

, (18)

f4(ξ) =R csc(

√Rξ)

r csc(√Rξ) + 1

, g4(ξ) =−√R cot(

√Rξ)

r csc(√Rξ) + 1

. (19)

Case 4: When R = r = 0

f5(ξ) =C

ξ= Cpg5(ξ), g5(ξ) =

1

pξ, (20)

where C is a constant.

Substitute the constants Ai, Bi, R, r and λi into Eq.(15) with along Eqs.(16)-(20) to obtain soliton

and periodic (or rational) solutions of the nonlinear PDE of concern. This method is used in chap-

ters 2, 3 and 4 of present thesis to solve nonlinear equations.

Apart from the above mentioned direct methods, several other direct and approximate methods

have also been used in the literature to solve nonlinear PDEs. For example, the Adomain decom-

position method [11], variational iteration method [11], homotopy analysis method [11], solitary

wave ansatz method [15], He’s variational method [16] and so on.

3 Work carried out

Here we present a brief account of the studies carried out on the analytic soliton solutions of some

important nonlinear physical models.

(1). The higher order nonlinear Schrodinger (HNLS) equationThe HNLS equation describes the propagation of femtosecond optical pulses in optical fibers and

can be written as [17]

iEx + a1Ett + a2|E|2E + i[a3Ettt + a4(|E|2E)t + a5E(|E|2)t] = 0. (21)

6

Here E represents the complex envelope of the electric field, the subscripts x and t are the nor-

malized spatial and temporal partial derivatives and a1, a2, a3, a4, a5 are respectively, the real pa-

rameters related to group velocity dispersion (GVD), self-phase modulation (SPM), third order

dispersion (TOD), self-steepening and self-frequency shift arising from stimulated Raman scat-

tering (SRS). This model, unlike the nonlinear Schrodinger equation (NLSE), is not integrable in

general. When the last three terms are omitted this propagation equation for the slowly varying

envelope of the electric field, E, reduces to the NLSE which is completely integrable by the inverse

scattering transform method [6]. The effect of third-order dispersion is significant for femtosecond

light pulses when the GVD is close to zero. It is negligible for optical pulses whose width is of the

order of 100fs or more, having power of the order of 1W and GVD far away from zero. However,

in this case self-steepening as well as self-frequency shift are still dominant and should be retained.

Here, we obtain the exact bright and dark soliton solutions for HNLS equation using the solitary

wave ansatz method. Restrictions on parameters of the soliton have been observed in course of the

derivation of soliton solutions.

Bright solitons are also known as bell-shaped solitons or non-topological solitons. These kinds of

solitons are modeled by the sech function. The intensity of the bright soliton solution takes the

form

|E(x, t)|2 =(2(k − a1ω

2 + a3ω3)

(a4ω − a2)

)cosh−2

[± 1

v

√k − a1ω2 + a3ω3

3a3ω − a1(x− vt)

], (22)

where k is the soliton frequency, ω is the soliton wave number.

The dark solitons are also known as topological solitons or simply topological defects. The intensity

of the dark soliton solution takes the form

|E(x, t)|2 =(3(3a3ω2v − 2 a1ωv + 1)

(2 a5 + 3 a4) v

)tanh2

[±

√(2a1ωv − 3a3ω2v − 1)

2a3v3(x− vt)

]. (23)

The existence of bright and dark soliton solutions given by Eqs.(22) and (23) depends on the specific

nonlinear and dispersive features of the medium, which have to satisfy the parametric constrained

conditions. From these expressions of soliton, we see that as the higher-order dispersive terms, i.e.

a1 and a3 increase, the amplitude of the solitary wave increases, while the pulse width gets narrower

and with increase in the value of self-frequency and stimulated Raman scattering i.e. increase in the

value of a4 and a5 , the amplitude of the solitary wave decreases, while the pulse width increases.

From the existence conditions of bright and dark solitons, we note that if the GVD and TOD are

both neglected i.e. a1 = a3 = 0, then both the bright and dark solitary waves disappear, but if TOD

exists i.e. a1 = 0 and a3 = 0, then the bright and dark solitary waves do not disappear.

We also found the explicit dark and bright soliton solutions of Eq.(21) by reducing it to quartic

anharmonic oscillator equation. These solitons appears as a special case of the periodic solutions

7

of anharmonic oscillator equation.

The dark solitary wave solutions is written as

E(x, t) =

√(k + a1ω2 − a3ω3)

(a4ω − a2)tanh

(√(k + a1ω2 − a3ω3)

2β2(3a3ω − a1)ξ

)exp[i(kx− ωt)]. (24)

From this soliton solution we also calculate the peak power and pulse width as P0 =(k+a1ω2−a3ω3)

(a4ω−a2)

and T0 =√

2β2(3a3ω−a1)(k+a1ω2−a3ω3)

. The formation condition of the dark solitary wave is k+a1ω2−a3ω3

β2(a1−3a3ω)< 0

and a4ω−a22β2(a1−3a3ω)

> 0.

The bright solitary wave solution is given by

E(x, t) =

√2(k + a1ω2 − a3ω3)

(a2 − a4ω)sech

(√(k + a1ω2 − a3ω3)

β2(a1 − 3a3ω)ξ

)exp[i(kx− ωt)]. (25)

The corresponding peak power and the pulse width are given as P0 = 2(k+a1ω2−a3ω3)(a2−a4ω)

and

T0 =√

β2(a1−3a3ω)(k+a1ω2−a3ω3)

. The formation condition of the bright solitary wave is k+a1ω2−a3ω3

β2(a1−3a3ω)> 0

and a4ω−a22β2(a1−3a3ω)

< 0. Note that the formation conditions of the bright and dark solitary waves are

opposite to each other. These obtained dark and bright solitary waves solutions propagate in the

normal (GVD > 0) as well as in anomalous dispersion regime (GVD < 0).

By using the 1-soliton solution, we have calculated a number of conserved quantities for Hirota and

Sasa-Satsuma cases. Finally, using He’s semi-inverse method, variational formulation was estab-

lished to obtain exact soliton solutions of HNLS equation.

(2). The HNLS equation with time-dependent coefficientsThe optical solitons in a Kerr law media is an important area of study which is governed by the

NLSE. It studies the propagation of solitons through optical fibers for trans-continental and tran-

oceanic distances. Thus the dynamics of solitons governed by the NLSE is well understood and well

known in this context. When inhomogeneities of the media and nonuniformity of the boundaries

are taken into account in various real physical situations, the variable-coefficient NLSE provides

more powerful and realistic models than their constant-coefficient counterparts. The governing en-

velope wave equation for femtosecond optical pulse propagation in inhomogeneous fiber takes the

time dependent form [18]

iEt + a1(t)Exx + a2(t)|E|2E + i[a3(t)Exxx + a4(t)(|E|2E)x + a5(t)E(|E|2)x] = 0. (26)

Here, the complex valued function E represents the wave profile where the independent variables

are the spatial x and time t and a1, a2, a3, a4, a5 are distributed parameters which are all time de-

pendent.

The intensity of topological soliton takes the form

|E(x, t)|2 = A20 tanh

2

[A0

√− a2(t) + a4(t)κ0

2a1(t) + 6a3(t)κ0

(x− v(t)t)

]. (27)

8

The time varying soliton parameters, the soliton frequency ω(t) and velocity v(t) have been cal-

culated during the course of derivation of solitons. Note that this solution exists provided that

constraint equation between the model coefficients a1(t), a2(t), a3(t), a4(t) and a5(t) is satisfied.

(3). Inhomogeneous NLSE with time-dependent coefficientsIt is well known that NLSE describes numerous nonlinear physical phenomena in the field of non-

linear science such as optical solitons in optical fibres, solitons in the mean-field theory of Bose-

Einstein condensates and the rogue waves (RWs) in the nonlinear oceanography etc. The oceanic

RWs can be, under the nonlinear theories of ocean waves, modeled by the dimensionless NLSE

iut +1

2uxx + |u|2u = 0, (28)

which describes the two-dimensional quasi-periodic deep-water trains in the lowest order in wave

steepness and spectral width. In the present work, we extend the NLSE (28) to the inhomogeneous

NLSE with variable coefficients, including group velocity dispersion β(t), linear potential V (x, t),

nonlinearity g(t) and the gain/loss term γ(t), in the form [19]

iut +β(t)

2uxx + V (x, t)u+ g(t)|u|2u = iγ(t)u, (29)

and found the bright and dark 1-soliton solutions using solitary wave ansatz under some parametric

restrictions.

(4). Variable coefficient NLSEThe varying dispersion and Kerr nonlinearity are of practical importance in a real optical-fiber

transmission system with the consideration of the inhomogeneities resulting from such factors as

the variation in the lattice parameters of the fiber media and fluctuation of the fiber’s diameters

[20]. Therefore, investigations on the variable-coefficient NLSE-type models for optical fibers

have become desirable.

Here, our emphasis is on the following variable-coefficient NLSE model [21]:

i[Ψz +

α(z)

2Ψ + σ(z)Ψt

]− 1

2β(z)Ψtt + γ(z)|Ψ|2Ψ = 0, (30)

where Ψ is a complex function of z and t. The function α(z) is the linear attenuation coefficient,

σ(z), β(z), and γ(z) are the inhomogeneous functions, respectively, related to the intermodal dis-

Table 1: Some single-JEF soliton solutionsSoliton type Soliton intensity ξ expression Existence condition

Case 1:Unchirped BS |Ψ′21|

2 = f220 exp

(−

∫ z0 α(z)dz

)sech2(ξ) ξ = p0t +

∫ z0 p0[βφ0 − σ]dz + q0

c4β(z)γ(z)

< 0

Case 2:Chirped BS |Ψ′22|

2 = f220 exp

[ ∫ z0

(κzκ

− α)dz

]sech2(ξ) ξ = p0κt + 1

2p0

∫ z0 κ[(φ0 − 2

∫ z0 σdz)κz − 2σ]dz + q0

c42γ(z)

dkdz

< 0

Case 3:Unchirped DS |Ψ′11|

2 = f220 exp

(−

∫ z0 α(z)dz

)tanh2(ξ) ξ = p0t +

∫ z0 p0[βφ0 − σ]dz + q0

c4β(z)γ(z)

> 0

Case 4:Chirped DS |Ψ′13|

2 = f220 exp

[ ∫ z0

(κzκ

− α)dz

]tanh2(ξ) ξ = p0κt + 1

2p0

∫ z0 κ[(φ0 − 2

∫ z0 σdz)κz − 2σ]dz + q0

c42γ(z)

dkdz

> 0

persion, GVD and nonlinear loss or gain. In practical applications, Eq.(30) and their various forms

9

are of considerable importance for the description of amplification, absorption, compression and

broadening of optical solitons in inhomogeneous optical fiber systems and also for the study of

stable transmission of solitons [22].

By using the F-expansion method, we construct exact solutions of the generalized NLSE with

varying intermodal dispersion and nonlinear gain or loss. Here we describe the dynamics of a few

analytic solutions which may be vital to improve the soliton transmission features in some actual

physical situations.

Table 1 depicts some interesting single-JEF soliton solutions. To display unique behavior of these

exact soliton solutions, one can choose the dispersion coefficient β(z) and the phase chirp parame-

ter k(z) in terms of trigonometric, hyperbolic functions, some constants and linear functions. Here,

we present some examples using the linear, trigonometric and hyperbolic distributed control sys-

tems.

From Table 1, note that the speed of chirpless bright soliton (BS) or dark soliton (DS) is related

to p0[β(z)φ0 − σ(z)] and phase shift is determined by∫ z

0[β(z)(φ2

0 − p20)− 2σ(z)φ0]dz. The wave

amplitude of BS is given by f20 exp(−12

∫ z

0α(z)dz) =

√−β(z)p20c4/γ(z), where c4β(z)

γ(z)< 0. So by

choosing suitable initial conditions and the intermodal dispersion and GVD parameters, the speed

and phase shift of unchirped bright and dark solitons in optical fiber communication systems can

be controlled. Similarly, in Table 1 we obtained different existence conditions for chirped and chirp

free soliton solutions. In unchirped case, the amplitude and solitary wave characteristics can exclu-

sively be controlled by the dispersion coefficients β(z) and the gain/loss coefficients γ(z). But for

solitary waves with chirp function κ(z) and the gain/loss coefficient γ(z) will determine the wave

propagation characteristics. To visualize the propagation characteristics of chirp and unchirped

dark-bright soliton solutions, few numerical simulations are given in the thesis.

(5). The (3+1)-dimensional NLSEThe pulse propagation in a cubic-quintic (CQ) isotropic dispersive medium is governed by the

generalized (3 + 1)D CQNLSE [23]

iΨz +1

2β(z)(∆⊥Ψ+Ψtt) + g3(z)|Ψ|2Ψ+ g5(z)|Ψ|4Ψ = iγ(z)Ψ, (31)

where Ψ(x, y, t; z) is the spatiotemporal field envelope propagating along the longitudinal distance

z, t is the retarded time, i.e., the time in the frame of reference moving with the wave packet

and ∆⊥ stands for the transverse Laplacian operator acting on spatial coordinates (x, y). The dis-

tributed parameters β, g3, g5 and γ are, respectively, the diffraction/dispersion coefficient, the cubic

and quintic nonlinearities and gain/loss coefficients.

We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)D

CQNLSE by using F-expansion technique. For restrictive parameters, these periodic wave solutions

acquire the form of localized spatial solitons. Such solutions exist under certain conditions, and im-

pose constraints on the functions describing dispersion, nonlinearity, and gain (or loss). We present

10

some cases of the periodic wave and light bullet soliton solutions, taking the diffraction (disper-

sion) coefficient β(z) of the form β(z) = β0 cos(kbz) and the gain (loss) coefficient γ(z) as a small

constant. This choice leads to alternating regions of positive and negative values of β(z), g3(z)

and g5(z), which is required for an eventual stability of soliton solutions. We then demonstrate the

nonlinear tunneling effects and controllable compression technique of 3-dimensional bright and

dark solitons when they pass unchanged through the potential barriers and wells affected by special

choices of the diffraction and/or the nonlinearity parameters. We show that when the bright soliton

(BS) passes through the diffraction barrier, its intensity grows and forms a peak near barrier, af-

terwards it again recovers its original shape while the intensity of dark soliton (DS) first decreases

and then increases in corresponding two asymptotic states near barrier and then it again attains its

original shape. Conversely, when the BS crosses the diffraction well, the intensity of the soliton

vanishes and forms a valley near well, while the intensity of DS first increases and then decreasing

in two asymptotic states near well respectively and after the tunneling, the solitons are restored

to their original shapes. For compression of soliton pulses, we consider a system with decaying

diffraction barrier and nonlinearity on an exponential background. It can be seen that after passing

the well the soliton is compressed about the position of well, which indicates that the pulse can be

compressed to a desired width and amplitude in a controllable (lossless, increasing or decreasing)

manner by the choice of the well (barrier) parameters. This indicates that a new pulse compression

technique might be developed for 3D soliton pulses. Direct numerical simulation has been per-

formed to show the stable propagation of bright soliton with 5% white noise perturbation.

(6). The (2+1)-dimensional NLSEIn this study, we consider the coupled (2+1)D nonlinear system of Schrodinger equations as

iut − uxx + uyy + |u|2u− 2uv = 0, (32)

vxx − vyy − (|u|2)xx = 0, (33)

where u(x, y, t) and v(x, y, t) are complex-valued functions. The (2+1)D NLSE play an vital role

in atomic physics, and the functions u and v have different physical meanings in different disci-

plines of physics [24, 25]. Main applications are, for instance, in fluid dynamics [24] and plasma

physics [25]. In the context of water waves, u is the amplitude of a surface wave packet while v

is the velocity potential of the mean flow interacting with the surface waves [25]. However, in the

hydrodynamic context, u is the envelope of the wave packet and v is the induced mean flow. In

addition, Eqs. (32) and (33) are relevant in a number of different physical contexts, describing slow

modulation effects of the complex amplitude v, due to a small nonlinearity, or a monochromatic

wave in a dispersive medium.

In the present work, we have reduced (2+1)D NLSE to a nonlinear ODE by using a simple trans-

formation then various solutions of the nonlinear ODE are obtained by using extended F-expansion

11

and projective Riccati equation methods. With the aid of solutions of the nonlinear ODE more ex-

plicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and

rational functions are found out.

(7). Density-independent diffusion reaction (DR) equationsWhile a variety of simplified versions of nonlinear DR equation are studied [26, 27, 28] in liter-

ature, but a DR equation with quadratic and cubic nonlinearities has been found useful in several

interesting applications recently [26, 27]. In fact, the presence of velocity-term in DR equations,

which is also attributed to the turbulent or anomalous diffusion in this later version is found to play

an important role in several applications, particularly in biological studies [26].

To obtain the exact solutions of the DR equations with quadratic and cubic nonlinearities, namely

Ct + vCx = DCxx + [a+ U(x)]C − β|C|C, (34)

Ct + vCx = DCxx + [a+ U(x)]C − β|C|2C, (35)

some simplifying assumptions are made, mainly to obtain the closed form solutions of Eqs.(34)

and (35) for complex concentration function C(x, t). Here, we relax this assumption of complex

C(x, t) and demonstrate that Eqs.(34) and (35) for real C(x, t) do admit a variety of exact solutions

including the solitonic ones in terms of hyperbolic functions. In particular, the existence of soliton

solutions of Eq.(35) are explicitly demonstrated in a certain parametric domain. We shall, however,

restrict ourselves to the case of constant value of the random potential U(x), say U0, in Eqs.(34)

and (35). In some sense Eqs.(34) and (35) can be considered as the generalization of the Fisher and

Nagumo equations respectively [26, 29]. For the solution of Eqs.(34) and (35), we introduce the

variable ζ = x− ωt and recast them for the real C(x, t) in the form

DC ′′ − (v − ω)C ′ + αC − βC2 = 0, (36)

DC ′′ − (v − ω)C ′ + αC − βC3 = 0, (37)

where U(x) is assumed to be constant (say U0) and α = (a+ U0).

Then, we apply the extended tanh-method for constructing a variety of exact traveling wave so-

lutions of Eqs.(34) and (35). The kink and antikink and some other traveling wave solutions are

expressed in terms of hyperbolic tangent functions. The kink and antikink soliton solutions of

Eqs.(34) and (35) are given by

C1(ζ) =α

4β

[1± tanh

(±√

α

24Dζ)]2

, (38)

C2(ζ) = ±√

α

4β

[1 + tanh

(±√

α

8Dζ)]

, (39)

respectively. Eqs.(38) and (39) have four traveling wave solutions corresponding to the two values

of µ and we label these solutions as C++(ζ), C+−(ζ), C−+(ζ) and C−−(ζ) corresponding to the

12

(upper, upper), (upper, lower), (lower, upper), (lower, lower) signs on the right side of these equa-

tions.

(8). Density-dependent DR equationsThere are various real physical phenomena in which, the diffusion coefficient D itself becomes

concentration or density dependent [26, 28]. For example, in certain insect dispersal model, D

depends on the population concentration C [26]. Thus, if concentration of a species increases then

its diffusion coefficient will also increase. In this work, we look for the exact solutions of the DR

equation with quadratic and quartic type nonlinearities along with a nonlinear convective flux term.

In particular, we investigate the solutions of following equations

Ct + kCCx = DCxx + αC − βC2, (40)

Ct + kC2Cx = DCxx + αC − βC4, (41)

where C = C(x, t) has the varying meaning depending on the phenomenon under study, D is

the diffusion coefficient and k, α and β are real constants. Eqs.(40) and (41) describe a trans-

port phenomenon in which both diffusion and convection processes are of equal importance i.e.,

the nonlinear diffusion could be thought of as equivalent to the nonlinear convection effects. In

particular, the second term on left hand side, kCCx (or kC2Cx) is the replacement of the conven-

tional vCx-term. It may be mentioned that the presence of vCx-type convective flux term in the DR

Eq.(35) makes the system nonconservative, whereas a nonlinear convection term in Eq.(40) or in

Eq.(41) arises as a natural extension of a conservation law. The soliton solutions of Eqs.(40) and

(41) are given by

C1(ζ) =α

2β

[1∓ tanh

(± αk

4βDζ)]

, (42)

C2(ζ) =βD

k2

[1± tanh

(± β2D

k3ζ)]

. (43)

For the positive value of k in Eq.(42), we represent these solutions as C−+(ζ) and C+−(ζ), which

in fact turn out to be kink soliton solutions and on the other hand negative value of k in Eq.(42),

represented by C−−(ζ) and C++(ζ) gives antikink soliton solutions.

(9). (2+1)-dimensional Maccari systemHere, we are interested to investigate new exact traveling wave solutions for the following (2 + 1)D

soliton equation

iut + uxx + uv = 0, (44a)

vt + vy + (uu∗)x = 0, (44b)

where u(x, y, t) is complex function and v(x, y, t) is real function. Eqs.(44a) and (44b) are similar

to integrable Zakharov equation in plasma physics to describe the behavior of sonic Langmuir soli-

tons, which are Langmuir oscillations trapped in regions of reduced plasma density caused by the

13

ponderomotive force due to a high-frequency field (when x = y in Eq. (44b). Recently Maccari

[30] obtained Eqs. (44a) and (44b) by an asymptotically exact reduction method based on Fourier

expansion and spatiotemporal rescaling from the Kadomtsev-Petviashvili equation. Maccari’s sys-

tem is a kind of nonlinear evolution equation that are often presented to describe the motion of the

isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma

physics, nonlinear optic, etc.

Similarly, like (2+1)D NLSE, we construct large number of traveling wave solutions of this system

using extended F-expansion and projective Ricatti equation methods [9].

(10). Modified KdV (mKdV) equation with time-dependent coefficientsThe mKdV equation appears in many fields such as acoustic waves in certain anharmonic lattices,

Alfven waves in a collisionless plasma, transmission lines in Schottky barrier, models of traffic con-

gestion, ion acoustic solitons, elastic media, etc.[6] Here we studied the following form of mKdV

equation with variable coefficients by using solitary wave ansatz:

ut + α(t)ux − β(t)u2ux + γ(t)uxxx = 0, (45)

where α(t), β(t) and γ(t) are all analytic functions of t. The bright soliton solution is given by

u(x, t) =

√−6γ(t)

β(t)B0 sech[B0(x− vt)], (46)

where the velocity v(t) is determined during integration of Eq.(45) and it should be noted that the

soliton solution Eq.(46) exists under the condition γ(t)β(t) < 0. Finally, the dark soliton solution

for the time-dependent mKdV Eq.(45) is written as

u(x, t) =

√6γ(t)

β(t)B0 tanh[B0(x− vt)], (47)

which shows that it is necessary to have β(t)γ(t) > 0 for soliton to exist.

(11). Generalized Gardner equation (GE)The generalized GE, with full nonlinearity is given by

(ul)t + [2α(t)um + 3β(t)u2m]ux + γ(t)(un)xxx = 0, (48)

where, l, m, and n are all positive real numbers. This generalization is based on the same kind as

given by the K(m,n) equation that is the generalization of the KdV equation. Hence, based on

the similarity of the structure, Eq.(48) will be referred to as the G(m,n) equation with generalized

evolution.

Similarly, like mKdV, we consider the 1-soliton solution of Eq.(48) of the form

u(x, t) =A

(λ+ cosh[B(x− vt)])p, (49)

14

where A = A(t), B = B(t) and v = v(t) are time-dependent coefficients which will be determined

as functions of the model coefficients α, β and γ. Here, in Eq.(49), A is the amplitude of the

soliton, while v is the velocity and B is the inverse width of the soliton. Here, in addition to the

usual soliton parameters, λ is a constant that will also be determined. The exponent p is unknown

at this stage, and its value will be obtained during the course of derivation of the 1- soliton solution

to the G(m,n) equation given by Eq.(48).

(12). Complex modified KdV equation with time dependent coefficientsThe variable coefficient complex mKdV equation, which reads as

ut + α(t)|u|2ux + β(t)uxxx = 0, (50)

where u is a complex valued function of the scaled horizontal coordinate x and time t, the sub-

scripts denote partial derivatives, and the analytic functions α(t) and β(t) stand for nonlinear and

dispersive effects, respectively. This model equation has been proposed as a model for the nonlinear

evolution of plasma waves [31, 32]. The physical model given by Eq.(50) describes the propaga-

tion of transverse waves in a molecular chain model and in a generalized elastic solid [31, 32]. The

two dimensional steady state distribution of lower-hybrid waves is governed by complex mKdV

equation.

In this work, we have derived bright and dark 1-soliton solutions of Eq.(50) of the form

u(x, t) =A

cosh[B(x− vt)]ei(−kx+ωt+θ), (51)

u(x, t) = A tanh[B(x− vt)]ei(−kx+ωt+θ), (52)

respectively. Here, A is the amplitude of the soliton, B is the inverse width and v is the soliton

velocity. Also k represents the frequency, ω is the wave number, and θ is the phase. It is to be noted

that since in Eq.(50), the coefficient of dispersion and nonlinearity are time dependent and may not

be constants. The soliton parameters A, B, k, ω and θ are time-dependent. We have obtained some

restrictions on parameters of the soliton while deriving soliton solutions.

3.1 Conclusions

In the introductory chapter one, a brief description of nonlinear phenomena and a survey on some

nonlinear models is presented. We have also discussed about solitons and various types of traveling

wave solutions. A brief account of some important and widely used analytic methods to obtain

exact solutions of a variety of nonlinear PDEs relevant to physical problems is also given.

In chapter 2, we have found exact bright, dark and cnoidal wave soliton solutions to the HNLS equa-

tion describing propagation of femtosecond light pulses in an optical fiber under certain parametric

conditions. Using soliton ansatz, both dark and bright solitons are constructed. The obtained solu-

tions may be useful to understand the mechanism of the complicated nonlinear physical phenomena

15

which are related to wave propagation in a higher order nonlinear Schrodinger model equation. It

is interesting to note that the existence of solitary wave solutions depends essentially on the model

coefficients, and therefore on the specific nonlinear and dispersive features of the medium. The

precise significance of these results may lie in the design of optical pulse compressors and solitary

wave based communication links. Integrals of motion for two specific cases of time independent

HNLS system are also obtained using exact 1-bright soliton solution. Similarly, by using solitary

wave ansatz the 1-soliton solutions of inhomogeneous NLSE with time dependent coefficients are

derived. Finally, we have shown the utility of the F-expansion technique to the generalized nonlin-

ear Schrodinger equation with varying parameters and successfully obtained a rich variety of abun-

dant exact solutions. These solutions include Jacobi elliptic function solutions, Weierstrass elliptic

function solutions, chirped and unchirped dark and bright solitary wave solutions and trigonometric

function solutions. The obtained solutions with arbitrary parameters may be significant to explain

some physical phenomena. All solutions are derived here with less algebraic expansion compu-

tations with the help of Maple. To visualize the dynamical behavior of spatiotemporal solitons,

numerical simulation for some solutions are given. While working with the F-expansion method

we have observed that if it is treated rigorously, the variable F-expansion method offers exact solu-

tions in a well-ordered form, from which one can construct solitary, trigonometric and Weierstrass-

elliptic function solutions. One of the drawbacks of the F-expansion method, by assuming the

solution of the equation in the polynomial form with many parameters, is that it sometimes leads

to inconsistent nonlinear algebraic systems.

In chapter 3, we have solved analytically the (3+1)D generalized CQNLSE with distributed diffrac-

tion, dispersion and gain/loss coefficients using the homogeneous balance principle and the F-

expansion technique. A number of exact periodic traveling wave and spatiotemporal solutions are

found. The tunneling effects of 3D dark and bright solitons through the diffraction barriers and

wells formed by the coefficients of diffraction and nonlinearity are investigated. It is also shown

that the soliton pulses can be compressed to a desired width and amplitude in a controllable manner

by the choice of the barrier (or well) parameters. The results obtained here, may be found potential

applications in areas such as nonlinear optical switches, all optical data processing schemes and in

the design of pulse compressors and amplifiers. Direct numerical simulation has been performed to

show the stable propagation of bright soliton with 5% white noise perturbation. The exact solution

of (2+1)D NLSE are obtained by using extended F-expansion and projective Riccati equation meth-

ods. Many different new forms of traveling wave solutions such as periodic wave solution, solitary

wave solution or bell-shaped soliton solutions, kink-shaped soliton solutions are obtained. These

results have proven that extended F-expansion method and projective Riccati equation method are

reliable and efficient in handling nonlinear problems.

In chapter 4, we gave exact solutions of density independent and dependent nonlinear diffusion

16

reaction equations.The kink and antikink and some other traveling wave solutions are expressed in

terms of hyperbolic tangent functions. In view of the fact, that nonlinear DR equations are in use

in explaining a variety of physical phenomena and the choice of nonlinearity in these equations

is a part of the modeling process of the phenomenon under study, it is expected that the results

obtained in this work can offer some clues in making such choices. The extended tanh method can

be applied, in principle, to any type of equation and the calculations are not so complicated as com-

pared to other methods. The extended tanh method is independent of the integrability of nonlinear

equations, so that it can be used to solve both integrable and non-integrable nonlinear evolution

equations. The bright and dark soliton solutions of variable coefficient modified KdV equation,

complex modified KdV equation and generalized Gardner equation are obtained by using solitary

wave ansatz method. Restrictions on parameters of the soliton have been observed in course of

the derivation of soliton solutions. Finally, a few 3D profiles of dark and bright solitons have been

given. We also investigated the Maccari equation by using extended F-expansion and projective

Riccati equation methods and obtained different kind of soliton solutions.

3.2 Future scope of the present study

The present study can further be extended on the following fronts:

• One may develop similarity transformation to the generalized (3+1)D CQNLSE, where it can

be reduced to the related constant coefficient (3+1)D CQNLSE and investigate the dynam-

ics of selfsimilar waves in dispersion decreasing fibers and dispersion changing periodically

fibers.

• One may generalized higher order HNLS equation by adding some higher order Kerr terms

or non-Kerr terms and find solitary wave solutions. This equation could be a model equation

of pulse propagation beyond ultrashort range in optical communication systems. These non-

Kerr terms are crucial when one increases the intensity of the incident light power to produce

shorter (femtosecond) pulses.

• The stability of soliton solutions is an important issue for practical problems. Thus stability

analysis of the obtained soliton solutions can be performed.

• The problems analytically solved here can be solved numerically. Such studies serve the

purpose of accuracy check of analytic results.

• One may generalized the (3+1)D NLSE with saturable nonlinearities and investigate the dy-

namical behavior of solitons.

17

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List of PublicationsResearch Papers in Journals:

1. “Analytical spatiotemporal soliton solutions to (3+1)-dimensional cubic-quintic nonlinear

Schrodinger equation with distributed coefficients”, Hitender Kumar, Anand Malik and

Fakir Chand, J. Math. Phys. 53, (2012) 103704.

2. “Exact solutions of nonlinear diffusion-reaction equations”, Anand Malik, Fakir Chand, Hi-tender Kumar and S.C. Mishra, Indian J. Phys. 86, (2012) 129.

3. “Exact solutions of nonlinear diffusion reaction equation with quadratic, cubic and quartic

nonlinearities”, Hitender Kumar, Anand Malik, Fakir Chand and S.C. Mishra, Indian J.

Phys. 86, (2012) 819.

4. “Exact solutions of some physical models using the (G′

G)-expansion method”, Anand Malik,

Fakir Chand, Hitender Kumar and S.C. Mishra, Pramana J. Phys. 78, (2012) 513.

5. “Exact traveling wave solutions of the Bogoyavlenskii equation using multiple (G′

G)-

expansion method”, Anand Malik, Hitender Kumar, Fakir Chand and S.C. Mishra, Comp.

Math. Appl. 64, (2012) 2850.

6. “Soliton solutions of some nonlinear evolution equations with time-dependent coefficients”,

Hitender Kumar, Anand Malik and Fakir Chand, Pramana J. Phys. 80, (2013) 361.

7. “Dark and bright solitary wave solutions of the higher order nonlinear Schrodinger equation

with self-steepening and self-frequency shift effects”, Hitender Kumar and Fakir Chand,

J. Nonl. Opt. Phys. Materials 22, (2013) 1350001.

8. “Applications of extended F-expansion and projective Ricatti equation methods to (2+1)-

dimensional soliton equations”, Hitender Kumar and Fakir Chand, AIP Advances 3, (2013)

032128.

9. “Optical solitary wave solutions for the higher order nonlinear Schrodinger equation with

self-steepening and self-frequency shift effects ”, Hitender Kumar and Fakir Chand, Opt.

Laser Tech. 54, (2013) 265.

10. “1-soliton solutions of complex modified KdV equation with time-dependent coefficients”,

Hitender Kumar and Fakir Chand, Indian J. Phys. (2013), DOI 10.1007/s12648-013-0310-

8.

11. “Chirped and chirpfree soliton solutions of generalized nonlinear Schrodinger equation with

variable coefficients ”, Hitender Kumar and Fakir Chand, (communicated).

20

Research Papers presented in Conferences/Workshops:

1. “ Exact soliton solutions of nonlinear diffusion reaction equation with quadratic and cubic

nonlinearities”, Hitender Kumar and Fakir Chand, “PNLD 2010”, held at IISc. Bangalore

from July 26-29, 2010.

2. “Exact traveling wave solutions of time-delayed and diffusion reaction nonlinear evolution

equations”, Hitender Kumar, Anand Malik and Fakir Chand, “Sixth NCNSD”, held at

Bharathidasan University, Tiruchirapalli from January, 27-30, 2011.

3. “Exact traveling wave solutions of the Bogoyavlenskii equation”, Anand Malik, HitenderKumar, Fakir Chand and S.C. Mishra, “19th IEEE workshop on Nonlinear Dynamics in

Electronic Systems”, held at IICB and SINP Kolkata from March, 8-11, 2011.

4. “Dark and Bright Solitary Wave Solutions of Higher Order Nonlinear Schrodinger Equation”,

Hitender Kumar, Anand Malik and Fakir Chand, “19th IEEE workshop on Nonlinear Dy-

namics in Electronic Systems”, held at IICB and SINP, Kolkata from March, 8-11, 2011.

5. “Analytic study of some Nonlinear Diffusion Reaction Equations”, Hitender Kumar, Anand

Malik and Fakir Chand, “19th IEEE workshop on Nonlinear Dynamics in Electronic Sys-

tems”, held at IICB and SINP, Kolkata from March, 8-11, 2011.

6. “Chirped and unchirped soliton solution of generalized nonlinear Schrodinger equation with

distributed coefficients”, Hitender Kumar and Fakir Chand, National Conference on Ad-

vances in Physics held at Department of Physics, IIT Roorkee from February 25-26, 2012.

21