34

CHAPTER 2

MATHEMATICAL METHODS

Research aimed at studying and solving problems inspired by

physics within a rigorous mathematical framework qualifies as a mathematical

method. Mathematical methods are extensively used and related to theoretical

physics and a lot of effort has gone on to put some physical theories into the

mathematical realm.

For example quantum mechanics and some aspects of functional

analysis parallel each other in many ways. The mathematical study of quantum

statistical mechanics has motivated results in operator algebra. The attempt to

construct a rigorous quantum field theory has brought about progress in fields

such as representation theory. Use of geometry and topology plays an

important role in string theory.

Thus it becomes imperative to use the tool of these mathematical

methods in order to understand the underlying physics behind the Bose-

Einstein condensates which are described by nonlinear partial differential

equations.

There are two kinds of mathematical methods to solve nonlinear

partial differential equations namely analytical method and numerical method.

35

2.1 ANALYTICAL METHODS

According to the definition of the word analyticus in Latin and

analutikos in Greek from which the word analytical is taken, it means

1 Dividing into elemental parts or basic principles

2 Reasoning or acting from a perception of the parts and

interrelations of a subject

3 Using analysis especially in thinking

4 Following logic necessarily

5 Involving algebra or other methods of mathematical analysis

6 Proving a known truth by reasoning from that which is to be

proved

What are the advantages of using an analytical method?

They provide precise solutions to the equation on hand and only

after obtaining an estimate of the analytical solution can we even think of a

numerical solution.

2.1.1 Introduction

There are several analytical methods to solve nonlinear partial

differential equations. All these soliton-possessing equations have a common

remarkable property i.e., they are completely integrable. Though the definition

of integrability is not well-defined, its general meaning can be taken as having

coherent structures that are unique. Scientists have identified some features

which might closely resolve the problem of determining the integrability of a

nonlinear partial differential equation. They are:

36

i) Inverse scattering transform

ii) Lax Pair and AKNS Scheme

iii) Hirota bilinear form

iv) Darboux Transformation

In this thesis work, Darboux transformation and Lax Pair

formulation are used to determine the integrability properties and to obtain

soliton solutions. In the subsequent sections, these analytical methods are

briefly discussed and their advantages and disadvantages are compared as

given by Lakshmanan & Rajasekar (2003).

2.1.2 Inverse Scattering Transform

Gardner et al (1967) solved the initial value problem of the KdV

equation through the method of Inverse scattering transform (IST). Later

Zakharov and Shabat (1972) showed that the same procedure could be

extended to nonlinear Schrodinger equation. The inverse scattering transform

is a nonlinear analogue of the Fourier transform method which has been

employed to solve several linear partial differential equations (PDEs).

Given the initial value of the potential )0,(xu and the boundary

conditions, one has to identify two linear differential operators L and B so that

we can convert the nonlinear partial differential equation into two linear

equations, namely a linear eigen value problem and a linear time evolution

equation as

L (2.1)

Bt (2.2)

37

The compatibility condition of the above two Equations (2.1) and

(2.2) gives

LBLt , (2.3)

The compatibility condition generates the nonlinear partial

differential equation one has started with. Once the linearization is performed

in the above sense for a given nonlinear dispersive system with the following

Equation (2.4)

)(uKu t (2.4)

where )(uK is a nonlinear functional of u and its spatial derivatives. The

Cauchy initial value problem corresponding to the boundary condition

0u as x can be solved by the following three step process and

indicated schematically and diagrammatically in Figure (2.1).

1. Direct Scattering transform analysis: Considering the initial

condition )0,(xu as the potential, an analysis of the linear eigen

value problem in Equation (2.1) is carried out to obtain the

scattering data S(0). For example, for the KdV equation

)0()0( nxS n=1,2,3,…,N, xxRCn )0,(),0( (2.5)

where N is the number of bound states with eigen values nx ,

Cn(0) is the normalization constant of the bound state eigen

functions and R(x,0) is the reflection coefficient for the

scattering data.

38

2. Time evolution of the scattering data: Using the asymptotic

form of the time evolution Equation (2.2) for the eigen

functions, the time evolution of the scattering data S(t) can be

determined.

3. Inverse scattering transform (IST) analysis: The set of Gelfand-

Levitan-Marchenko integral equation corresponding to the

scattering data S(t) is constructed and solved. The resulting

solution consists typically of N number of localized and

exponentially decaying asymptotic solutions. In this way one

can successfully solve the initial value problem of the soliton

equations.

From the above three steps it is clear that solving the initial value

problem of the given nonlinear partial differential equation boils down to

solving an integral equation. In this perspective, generating soliton solutions

using the inverse scattering transform method is quite complicated and

intricate.

39

Figure 2.1 Schematic diagram of the inverse scattering transform

method

2.1.3 Lax Pair and AKNS Scheme

Lax (1968) introduced a novel technique to obtain soliton solutions

of PDEs using a matrix formalism, in which an NLPDE is expressed as a

compatibility conditions of two linear equations for a wave function

),,( tx as:

u(x,0)

u(x,t)

Scattering

data S(0)

at time tDirect Scattering

Scattering

data S(t)at time t

Inverse Scattering

Time

Evolution

Of

Scattering

data

40

L (2.6)

Bt (2.7)

where L and B are differential operatiors in the t derivatives. The pair (L,B) is

called the Lax pair of the integrable system. Equation (2.6) gives the eigen

value problem with the eigen value and Equation (2.7) determines the t

evolution of the wave function . The eigen value is considered to be

invariant under the t-evolution. i.e.,

0dt

d (2.8)

The Lax operator L evolves in such a way that its spectrum remains

constant and hence it is known as an isospectral problem. The invariance of

is the reason for the robustness of the soliton and therefore is the most

important property in the application to a soliton communication system.

To obtain the given nonlinear partial differential equation from the

Lax pair, one has to impose a condition known as a compatibility condition as

there are two equations for a single function . The requirement in Equation

(2.8) leads to such a compatibility condition by taking the x-derivation of

Equation (2.6) and then using Equation (2.7) as follows:

LBLBBLt

L, (2.9)

Equation (2.9) is called as the Lax equation and gives the operator

representation or Lax formalism of an integrable system. Lax used this

formalism to solve the KdV equation for which,

41

qz

L2

2

3

3

463zz

qqzB (2.10)

Therefore if a nonlinear partial differential equation arises as the

compatibility condition of two such operators L and B, then Equation (2.9) is

called the Lax representation of the partial differential equation and L and B

are called as the Lax pair.

One may say that the eigenvalue problem is isospectral. The Lax

condition in Equation (2.8) is the isospectral condition for the Lax pair L and

B. In mathematics, two linear operators are called isospectral or cospectral if

they have the same spectrum. Roughly speaking, they are supposed to have the

same sets of eigenvalues, when those are counted with multiplicity.

The theory of isospectral operators is markedly different depending

on whether the space is finite or infinite dimensional. In finite-dimensions, one

essentially deals with square matrices.

In the case of operators on finite-dimensional vector spaces,

for complex square matrices, the relation of being isospectral for two

diagonalizable matrices is just similarity. This doesn't however reduce

completely the interest of the concept, since we can have anisospectral

family of matrices of shape A(t) = M(t)1AM(t) depending on a parameter t in a

complicated way. This is an evolution of a matrix that happens inside one

similarity class.

A fundamental insight in soliton theory was that the infinitesimal

analogue of the preservation of spectrum was an interpretation of the

conservation mechanism. The identification of so-called Lax pairs (L,B) giving

42

rise to analogous equations, by Peter Lax, showed how linear machinery could

explain the non-linear behaviour.

Ablowitz, Kaup, Newell and Segur (1974) extended this Lax

formalism to solve a wider class of NLPDEs such as modified KdV, sine-

Gordon and NLS equations. This method is now-a-days known as AKNS

formalism. They considered two linear equations of the form:

Lx

Bt where T

21 , (2.11)

where is a n-dimensional vector and L and B are (n x n) matrices. If one

requires Equation (2.11) to be compatible, then it requires txxt and L and

B must satisfy

0, BLBL xt (2.12)

Equation (2.12) is more general than that given by Lax as it allows

eigenvalue dependence other than L . As an example, the linear eigen

value problem for optical solitons in NLS system can be constructed with L

and B given in the form:

2/

2/*

iq

qiL and

PO

NMB (2.13)

where is the eigen value parameter. To determine the values of M, N, O and

P the following expansions are used:

43

2

2

10 MMMM

2

2

10 NNNN (2.14)

And so on. Then, using the compatibility condition (2.12), B matrix

is obtained as:

2

2

2

0*

0

20

02

qiiq

iqqi

q

q

i

i

Bx

x (2.15)

The compatibility condition 0, BLBL xt gives the nonlinear

Schrodinger equation for bright solitons of the form:

022qqqiq xxt (2.16)

It is interesting to note that the Ablowitz, Kaup, Newell and Segur

(AKNS) formalism can be extended to inhomogeneous systems by taking the

spectral parameter as a function of x and t.

2.1.4 Hirota bilinear form

Although the inverse scattering transform method was the first

analytical technique developed to solve nonlinear partial differential equations,

it was complicated and intricate as it involved solving integral equations.

Moreover one should have prior knowledge of the initial data and the

boundary conditions imposed on it. On the other hand, the Darboux

transformation is iterative in nature and uses simple algebra without involving

complex mathematics but it warrants the identification of the Lax-pair of the

associated dynamical system.

44

Hirota’s (1971) bilinearisation method does not need any prior

information about the potential or the physical field associated with the

nonlinear partial differential equation or even the Lax pair of the associated

dynamical system. Hirota’s method has an inbuilt algebraic and geometric

structure, is more elegant and straightforward and can be directly employed to

generate soliton solutions of nonlinear partial differential equations. The

salient features of the Hirota method are the following

1. The given nonlinear partial differential equation has to be

converted into a bilinear equation through a transformation

which can be identified from the Painleve analysis. Each term

of the bilinear equation has the degree two.

2. The dependent variables in the bilinear form have to be

expanded in the form of a power series in terms of a small

parameter.

3. After substituting the dependent variables into the bilinear

form and equating the different powers of the small parameter,

a set of linear partial differential equations can be generated.

4. Finally, solving the linear partial differential equations, one can

generate the soliton solutions.

The key success of the Hirota method lies in the identification of the

dependent variable transformation as well as in choosing an optimum power

series to linearise the given nonlinear partial differential equation. In recent

times, Hirota’s method has been profitably used to generate exponentially

localized structures called dromions as explained by Radha and Lakshmanan

(1994) and more useful for systems whose Lax pair is not yet known.

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2.1.5 Darboux Transformation

Darboux transformations, originally introduced by Darboux (1882)

in a theorem on second-order differential equations, represent a powerful tool

in generating families of exactly solvable Hamiltonians. They allow controlled

manipulations of the spectrum and are therefore closely related with

supersymmetric quantum mechanics and inverse problems in quantum

scattering theory. Today Darboux transformation is at least of equal

importance because applications of inverse scattering techniques to

experimental data are almost exclusively based on them.

Darboux transformations, which are directly related to Backlund

transformations, have also become an essential ingredient in the study of

nonlinear partial differential equations. The direct search for exact solutions to

nonlinear partial differential equations has become more and more attractive

partly due to the availability of computer symbolic systems like Maple or

Mathematica, which allows one to perform some complicated and tedious

algebraic calculation on computer, and helps to find and plot new exact

solutions to the Partial Differential Equations.

Darboux Transformations are done in order to solve nonlinear

partial differential equations when their lax pair-a system of two linear

equations used instead of the nonlinear partial differential equation.

Advantages of using a Darboux Transformation over other methods to solve a

nonlinear partial differential equation is as follows

1 Darboux Transformation is simple and straight forward

2 Darboux Transformation algorithm is a purely algebraic one

3 The scattering data of the system need not be known

4 The Lax pair of the system should be known

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2.1.5.1 How is a Darboux Transformation done?

According to Khawaja (2009), the following are the steps involved

in making a Darboux Transformation

1. Applying the Darboux transformation method on nonlinear

partial differential equations requires finding a linear system of

equations for an auxiliary field ),( tx . The linear system is

usually written in a compact form in terms of a pair of matrices

as follows:

.Ux and .Vt . The matrices U and V, known as the

Lax pair, are functionals of the solution of the differential

equation.

2. The consistency condition of the linear system txxt is

required to be equivalent to the partial differential equation

under consideration. Consistency condition can also be called

the compatibility condition when it involves the matrices as

0,VUVU xt

In the resulting matrix the diagonal elements are zero and the

off diagonal elements give us the nonlinear partial differential

equation under consideration. We can then conclude that

The lax pair is correct for the given nonlinear partial

differential equation

The lax pair or the set of two linear equations can be used

to linearise the nonlinear partial differential equation.

3. Applying the Darboux transformation, as defined below, on

transforms it into another field ]1[ .

47

)(]1[ SI where

1HHS

12

21H1

1

0

0

The S Matrix elements become

lkklklS

)( 111

For the transformed field ]1[ to be a solution of the linear

system, the Lax pair must also be transformed in a certain

manner.

4. The transformed Lax pair will be a functional of a new

solution of the same differential equation. Practically, this is

performed as follows. First, we find the Lax pair and an exact

solution of the differential equation, known as the seed

solution. Fortunately, the trivial solution can be used as a seed,

leading to nontrivial solutions.

The lax equation with a seed solution q=0 becomes

.0Ux and .0Vt .

Using the Lax pair and the seed solution, the linear system is

then solved and the components of are found. The new

solution is expressed in terms of these components and the

seed solution.

From the first lax equation of .0Ux we get a solution of

)(x

48

From the second lax equation of .0Vt we get a solution

of )(t

But we know that our )()(),( txtx

5. It should be emphasized that while applying the Darboux

transformation is almost straight forward, finding a linear

system that corresponds to the differential equation at hand is

certainly not a trivial matter. Usually, this is found by trial and

error, or by starting from a certain linear system and then

finding the differential equation it corresponds to. Khawaja has

introduced a systematic approach to find the linear system

described briefly.

6. The partial derivatives of the auxiliary field, x and t , are

expanded in powers of with unknown matrix coefficients.

The expansions are terminated at the first order for x and the

second order for t since this will be sufficient to generate the

class of Gross– Pitaevskii equations under consideration. The

higher order matrix coefficients turn out to be essentially

determined by the zeroth-order matrix coefficients of U and V.

7. To find the matrices U and V, we expand them in powers of the

wavefunction ),( tx , its complex conjugate, and their partial

derivatives. The coefficients of the expansions are unknown

functions of x and t. Substituting these expansions in the

consistency condition, we find a set of equations for the

unknown function coefficients.

8. Finally, by solving these equations the Lax pair and

consequently the linear system will be determined. The linear

system found here is a generalization to that of Zakharov–

49

Shabat for homogeneous Gross–Pitaevskii equation. The

consistency condition leads to the following compatibility

relation between the matrices U and V:

0,VUVU xt ,

where UVVUVU .., is called the commutator of matrices U

and V which are nothing but quantum mechanical operators.

2.2 NUMERICAL METHODS

2.2.1 Introduction

According to Sastry (2004), partial differential equations occur in

many branches of applied mathematics, for example in hydrodynamics,

elasticity, quantum mechanics and electromagnetic theory. The analytical

treatment of these equations is a rather involved process and requires

application of advanced mathematical methods.

On the other hand, it is generally easier to produce sufficiently

approximate solutions by simple and efficient numerical methods. Solution of

partial differential equations by numerical methods in general has the

following advantages over analytical solutions:

1. The equations used in numerical techniques are much more

intuitive. Students can clearly understand the meaning of a

numerical equation and can easily generate various values of

the function by hand or by using Excel. The exponential form

of the analytical solution is clear to those with strong

mathematics skills but not so clear to others.

2. The basic procedure of evolution of a numerical technique

)()()( SdtSdttS is the same regardless of how

50

complicated the formulas are which describe d(S). This is not

true of analytical solutions as it is relatively easy to get into

mathematics which is much too complicated to obtain

analytical solutions. Thus more realistic models of greater

complexity can be investigated using numerical techniques.

3. By the use of numerical methods, a majority of phenomena and

processes can be taken into account without the principle

problems and therefore a more exact design or accurate

optimizing calculations are possible.

4. Some phenomena and processes can be realized only with great

difficulty or with respect to a destruction of the tested device

which could be very expensive.

5. Possibility to simulate these phenomena and processes is one

of the greatest advantage of numerical methods

6. However, a correct physical interpretation and determination of

input boundary conditions and of material properties are

necessary for a successful solution of numerical models.

There are two kinds of numerical methods for the solution of partial

differential equations, namely the finite difference and finite-element

approaches. The finite difference methods are conceptually easier to

understand and are easy to program for systems that can be approximated with

uniform grids. However, they are difficult to apply to systems with

complicated geometries. Although the finite element method is based on some

fairly straightforward ideas, the mechanics of generating a good finite-element

code for two and three dimensional problems is not a trivial exercise. It is also

computationally expensive for larger problems. However, it is vastly superior

to the finite difference approaches for systems involving complicated shapes.

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2.2.2 Methods for Numerical Solution

There are several methods for the numerical solution of the

nonlinear partial differential equation in general. Fourier methods and finite

difference methods are the broad classification of these numerical methods.

With the advent of the Fast Fourier Transform (FFT), a fundamental problem-

solving tool, Fourier methods could be applied effectively without

sophisticated training or years of experience. Most of the finite difference

methods consist of three basic steps namely

1 Division of spatial domain into an orthogonal computational

grid

2 Discretization of the governing equations and boundary

conditions in space and time to derive approximately equivalent

algebraic equations for each node.

3 Solving the resulting equations by a suitable matrix inversion or

iterative technique

2.2.2.1 Crank-Nicolson method

The Crank–Nicolson method is a finite difference method used for

numerically solving the heat equation and similar partial differential equations.

It is a second-order method in time, it is implicit in time and can be written as

an implicit Runge–Kutta method, and it is numerically stable. The method was

developed by John Crank (1947) and Phyllis Nicolson in the mid 20th century.

For diffusion equations (and many other equations), it can be shown

the Crank–Nicolson method is unconditionally stable. However, the

approximate solutions can still contain (decaying) spurious oscillations if the

ratio of time step t times the thermal diffusivity to the square of space step,

x2, is large (typically larger than ½)

52

We try to solve the GP equation using the Crank Nicholson method

which is a finite difference method. The algorithm of the Crank-Nicolson

method is as follows:

1. Suppose we wish to find ),( txu satisfying the PDE

0txxx ucubuau

Subject to the initial condition )()0,( xfxu

2. One cannot calculate the entire function u instead we shall

consider the solution as the numerical values that u takes on a

grid of x,t values placed over some domain of interest.

3. Suppose we have a rectangular domain with

x ranging from minx to maxx

t ranging from 0 to T

4. Divide maxmin , xx into I equally spaced intervals at x values

indexed by Ii ...1,0 . Divide T,0 into N equally spaced

intervals at t values indexed by Nn ,...1,0

The length of the interval is k in the t direction and h in the x

direction.

5. We seek an approximation to the true values of u at the

11 IXN grid points. Let niu , denote our grid point where

ihxx min and nkt

53

Figure 2.2 A mesh of x and t having I X N intervals

7. The next step which makes the procedure a finite difference

method is to approximate the partial derivatives of u at each

grid point by the finite difference formulas. Different finite

difference methods use different finite difference formulas to

denote the partial derivatives of u. The Crank-Nicolson method

is approximated by replacing time with the backward

difference approximation and space with the central difference

approximation.

8. One could proceed to calculate all the 1,niu from the niu , and

recursively obtain u for the entire grid.

54

9. The result of this is called an explicit finite difference solution

for u. It is second order accurate in the x direction, though

only first order accurate in the t direction, and easy to

implement. Unfortunately the numerical solution is unstable

unless the ratio 2/ hk is sufficiently small. Thus when a direct

computation of the dependent variables can be made in terms

of known quantities, the computation is said to be explicit.

10. When the dependent variables are defined by coupled sets of

equations, and either a matrix or iterative technique is needed

to obtain the solution, the numerical method is implicit.

11. The consequences of using an implicit Vs explicit solution for

a time dependent problem depends on two parts namely

numerical stability and numerical accuracy. The instability

problem can be handled by using an implicit finite difference

scheme. This is the recommended method for most problems

in the Crank-Nicolson algorithm, which has the virtues of

being unconditionally stable (i.e. for all 2/ hk ) and also is

second order accurate in both the x and t directions. Thus the

principal reason for using implicit methods which are more

complex to program and require more complicated effort in

each solution step is to allow for large time-step sizes.

2.2.2.2 Split-Step Crank-Nicolson method

Adhikari and Muruganandham (2010) have used a combination of a

Fourier method namely Split-Step Fourier method and a finite difference

method namely the Crank-Nicolson Finite difference method in their new

numerical method called the Split-Step Crank-Nicolson (SSCN) method. It

becomes worthwhile to understand the salient features of the SSCN method

55

The GPE can be written in the operator form as

Ht

i (2.17)

where the Hamiltonian H contains the different nonlinear and linear terms

including the spatial derivatives. In the split step Crank-Nicholson Method, the

iteration is done in several steps by breaking up the full Hamiltonian into

different derivative and non-derivative parts. So 321 HHHH where

2

1 xH (2.18)

2

2

2x

H (2.19)

13 HH (2.20)

The time variable is discretized as ntn where is the time step.

The solution is advanced first over the time step at time nt by solving the

GPE with 1HH to produce the first intermediate solution and from this we

generate the second intermediate solution by following semi-implicit Crank-

Nicholson scheme and then obtain the final solution.

As there is no derivative term in H1 this propagation is performed

essentially exactly for small through the operation

nHi

nnd

n eH 1)( 1

3/1 (2.21)

where )( 1Hnd denotes time-evolution operation with H1 and the suffix ‘nd’

denotes the non-derivative. Next we perform the time propagation

corresponding to the operator H2 numerically by following semi-implicit

Crank-Nicolson scheme

3/13/2

2

3/13/2

2

1 nnnn

Hi

(2.22)

56

The formal solution to Equation (2.22) is

3/1

2

23/1

2

3/2

2/1

2/1)( nn

CN

n

Hi

HiH (2.23)

where CN denotes the time-evolution operation with 2H and the suffix ‘CN’

refers to Crank-Nicholson algorithm. Operation CN is used to propagate the

intermediate solution 3/1n by the time step to generate the second

intermediate solution 3/2n . The final solution is obtained from

n

ndCNnd

n HHH )()()( 123

1 (2.24)

The break-up of the non- derivative term in two parts 1H and

3H symmetrically around the derivative term 2H , increases enormously the

stability of the method and reduces the numerical error.

The advantage of the above split-step method with small time step is

due to the following three factors.

1 First, all iterations conserve normalization of the wave function.

2 Second, the error involved in splitting the Hamiltonian is proportional

to 2 and can be neglected and the method preserves the symplectic

structure of the Hamiltonian formulation.

3 Finally, as a major part of the Hamiltonian including the

nonlinear term is treated fairly accurately without mixing

with the delicate Crank-Nicolson propagation, the method

can deal with an arbitrarily large nonlinear term and lead to

stable and accurate converged result.