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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.
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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:
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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)
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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.
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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.
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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
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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,
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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
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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:
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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.
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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[ .
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)(]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
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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–
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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
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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)
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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
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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.
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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
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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)
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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.