## Abstract

We implement the unified transform method to the initial-boundary value (IBV) problem of the Sasa–Satsuma equation on the half line. In addition to presenting the basic Riemann–Hilbert formalism, which linearizes this IBV problem, we also analyse the associated general Dirichlet to Neumann map using the so-called global relation.

## 1. Introduction

Several of the most important PDEs in mathematics and physics are integrable (in this paper, integrable means the PDEs admit Lax pair). Integrable PDEs can be analysed by means of the inverse scattering transform (IST) formalism. Until the 1990s, the IST methodology was pursued almost entirely for pure initial value problems. However, in many laboratory and field situations, the wave motion is initiated by what corresponds to the imposition of boundary conditions rather than initial conditions. This naturally leads to the formulation of initial-boundary value (IBV) problems instead of a pure initial value problem.

In 1997, Fokas announced a new unified approach for the analysis of IBV problems for linear and nonlinear integrable PDEs [1,2] (see also [3]). The Fokas method provides a generalization of the IST formalism from initial value to IBV problems, and over the last 15 years, this method has been used to analyse boundary value problems for several of the most important integrable equations possessing 2×2 Lax pairs, such as the Korteweg–de Vries, the nonlinear Schrödinger (NLS), the sine-Gordon and the stationary axisymmetric Einstein equations, e.g. [4–11]. Just like the IST on the line, the unified method yields an expression for the solution of an IBV problem in terms of the solution of a Riemann–Hilbert problem. In particular, the asymptotic behaviour of the solution can be analysed in an effective way by using this Riemann–Hilbert problem and by employing the nonlinear version of the steepest descent method introduced by Deift & Zhou [12]. However, for the formulation of this Riemann–Hilbert problem, both the *x*- and the *t*-parts of the Lax pair play an important role. Actually, it is the *t*-part which determines where, in the complex *k*-plane, the jumps occur. It differs significantly from the IST formalism because it just needs the *x*-part of the Lax pair to formulate a Riemann–Hilbert problem, the *t*-part will just be used to determine the time evolution of scattering data.

It is well known that the NLS equation 1.1describes slowly varying wave envelopes in dispersive media and arises in various physical systems including water waves, plasma physics, solid-state physics and nonlinear optics. One of the most successful related applications is the description of optical solitons in fibres. However, several phenomena have been observed by experimental which cannot be explained by equation (1.1). In order to understand such phenomena, Kodama and Hasegawa proposed the higher order NLS equation 1.2

In general, equation (1.2) is not completely integrable, unless certain restrictions are imposed on the real parameters *β*_{1}, *β*_{2} and *β*_{3}. In particular, the following four cases, including the NLS equation, are known to be solvable:

— the derivative NLS equation-type I (

*β*_{1}:*β*_{2}:*β*_{3}=0:1:1),— the derivative NLS equation-type II (

*β*_{1}:*β*_{2}:*β*_{3}=0:1:0),— the Hirota equation (

*β*_{1}:*β*_{2}:*β*_{3}=1:6:0),— the Sasa–Satsuma (SS) equation (

*β*_{1}:*β*_{2}:*β*_{3}=1:6:3), 1.3

Recently, Lenells [13] implemented the Fokas method to IBV problems for integrable evolution equations with Lax pairs involving 3×3 matrices. He also used this method to analyse the Degasperis–Procesi eqn in [14]. In this paper, following this approach, we analyse the IBV problem of the SS equation on the half-line by using this method. The IST formalism for the initial value problem of the SS equation has been obtained in [15].

According to [15], we introduce variable transformations, 1.4a 1.4b and 1.4cThen equation (1.2) reduces to a complex modified KdV-type equation 1.5

### (a) Organization of the paper

In §2, we perform the spectral analysis of the associated Lax pair. We formulate the main Riemann–Hilbert problem in §3 and we analyse the associated general Dirichlet to Neumann map using the so-called global relation in §4.

## 2. Spectral analysis

The Lax pair of equation (1.5) is [15]
2.1aand
2.1bwhere
2.2and
2.3where
2.4with
2.5In the following, we let *ε*=1 for the convenience of the analysis.

### (a) The closed one-form

Suppose that *u*(*x*,*t*) is a sufficiently smooth function of (*x*,*t*) in the half-line domain which decay as . Introducing a new eigenfunction *μ*(*x*,*t*,*k*) by
2.6we then find the Lax pair equations
2.7Equations (2.7) can be written in differential form as
2.8where *W*(*x*,*t*,*k*) is the closed one-form defined by
2.9where denotes the operator, which acts on a 3×3 matrix *X* by .

### (b) The *μ*_{j}'s

We define three eigenfunctions of (2.7) by the Volterra integral equations
2.10where *W*_{j} is given by (2.9) with *μ* replaced with *μ*_{j}, and the contours are shown in figure 1. The first, second and third column of matrix equation (2.10) involves the exponentials
2.11We have the following inequalities on the contours:
2.12So, these inequalities imply that the functions are bounded and analytical for such that *k* belongs to
2.13where denote four open, pairwise disjointed subsets of the Riemann *k*-sphere shown in figure 2. And the sets have the following properties:
where *l*_{i}(*k*) and *z*_{i}(*k*) are the diagonal entries of matrices −i*kΛ* and −4i*k*^{3}*Λ*, respectively.

In fact, for *x*=0, *μ*_{1}(0,*t*,*k*) has enlarged the domain of boundedness: (*D*_{2}∪*D*_{4},*D*_{2}∪*D*_{4},*D*_{1}∪*D*_{3}) and *μ*_{2}(0,*t*,*k*) have enlarged the domain of boundedness: (*D*_{1}∪*D*_{3},*D*_{1}∪*D*_{3},*D*_{2}∪*D*_{4}).

### (c) The *M*_{n}'s

For each *n*=1,…,4, a solution *M*_{n}(*x*,*t*,*k*) of (2.7) is defined by the following system of integral equations:
2.14where *W*_{n} is given by (2.9) with *μ* replaced with *M*_{n}, and the contours , *n*=1,…,4, *i*,*j*=1,2,3 are defined by
2.15According to the definition of the *γ*^{n}, we find that
2.16

The following proposition ascertains that the *M*_{n}'s defined in this way have the properties required for the formulation of a Riemann–Hilbert problem.

### Proposition 2.1

*For each n*=1,…,4, *the function M*_{n}(*x*,*t*,*k*) *is well defined by equation* (2.14) *for* *and* (*x*,*t*)∈*Ω*. *For any fixed point* (*x*,*t*), *M*_{n} *is bounded and analytical as a function of k*∈*D*_{n} *away from a possible discrete set of singularities* {*k*_{j}} *at which the Fredholm determinant vanishes. Moreover, M*_{n} *admits a bounded and continuous extension to* *and*
2.17

### Proof.

The boundedness and analyticity properties are established in appendix B in [13]. Substituting the expansion
into the Lax pair (2.7) and comparing the terms of the same order of *k* yields equation (2.17). □

### Remark 2.2

We have defined two sets of eigenfunctions: and . The unified approach of Fokas [1] for Lax pairs involving 2×2 matrices also implicitly uses two types of eigenfunctions: the *μ*_{j}'s are used for the spectral analysis, whereas the Riemann–Hilbert problem is formulated in terms of another set of eigenfunctions; our *M*_{n}'s are the analogues of this latter set of eigenfunctions, see §2*d*.

### (d) The jump matrices

We define spectral functions *S*_{n}(*k*), *n*=1,…,4, and
2.18Let *M* denote the sectionally analytical function on the Riemann *k*-sphere which equals *M*_{n} for *k*∈*D*_{n}. Then, *M* satisfies the jump conditions
2.19where the jump matrices *J*_{m,n}(*x*,*t*,*k*) are defined by
2.20

### Remark 2.3

As the integral equations (2.14) that define *M*_{n}(0,0,*k*) involve only integration along the initial half-line and along the boundary {*x*=0,0<*t*<*T*}, the *S*_{n}'s (and hence also the *J*_{m,n}'s) can be computed from the initial and boundary data alone. Thus, relation (2.19) provides the jump condition for a Riemann–Hilbert problem, which, in the absence of singularities, can be used to reconstruct the solution *u*(*x*,*t*) from the initial and boundary data. However, if the *M*_{n}'s have pole singularities at some points {*k*_{j}}, , the Riemann–Hilbert problem needs to include the residue conditions at these points. For the purpose of determining the correct residue conditions (and also for the purposes of analysing the nonlinearizable boundary conditions in §4), it is convenient to introduce three eigenfunctions in addition to the *M*_{n}'s.

### (e) The adjugated eigenfunctions

We will also need the analyticity and boundedness properties of the minors of the matrices . We recall that the cofactor matrix *X*^{A} of a 3×3 matrix *X* is defined by
where *m*_{ij}(*X*) denote the (*ij*)th minor of *X*.

It follows from (2.7) that the adjugated eigenfunction *μ*^{A} satisfies the Lax pair
2.21where *V* ^{T} denotes the transform of a matrix *V* . Thus, the eigenfunctions are solutions of the integral equations
2.22Then we can get the following analyticity and boundedness properties:
2.23In fact, for *x*=0, has enlarged the domain of boundedness: (*D*_{1}∪*D*_{3},*D*_{1}∪*D*_{3},*D*_{2}∪*D*_{4}) and have enlarged the domain of boundedness: (*D*_{2}∪*D*_{4},*D*_{2}∪*D*_{4},*D*_{1}∪*D*_{3}).

### (f) The *J*_{m,n}'s computation

Let us define the 3×3 matrix value spectral functions *s*(*k*) and *S*(*k*) by
2.24aand
2.24bThus,
2.25We deduce from the properties of *μ*_{j} and that *s*(*k*) and *S*(*k*) have the following boundedness properties:
Moreover,
2.26

### Proposition 2.4

*The S*_{n} *can be expressed in terms of the entries of s*(*k*) *and S*(*k*) *as follows*:
2.27a*and*
2.27b

### Proof.

Let denote the contour in the (*x*,*t*)-plane, here *X*_{0}>0 is a constant. We introduce *μ*_{3}(*x*,*t*,*k*;*X*_{0}) as the solution of (2.10) with *j*=3 and with the contour *γ*_{3} replaced by . Similarly, we define *M*_{n}(*x*,*t*,*k*;*X*_{0}) as the solution of (2.14) with *γ*_{3} replaced by . We will first derive expression for *S*_{n}(*k*;*X*_{0})=*M*_{n}(0,0,*k*;*X*_{0}) in terms of *S*(*k*) and *s*(*k*;*X*_{0})=*μ*_{3}(0,0,*k*;*X*_{0}). Then (2.27) will follow by taking the limit .

First, we have the following relations:
2.28Then we get *R*_{n}(*k*;*X*_{0}) and *T*_{n}(*k*;*X*_{0}) are defined as follows:
2.29aand
2.29bRelations (2.28) imply that
2.30These equations constitute a matrix factorization problem which, given {*s*,*S*} can be solved for the {*R*_{n},*S*_{n},*T*_{n}}. Indeed, integral equations (2.14) together with the definitions of {*R*_{n},*S*_{n},*T*_{n}} imply that
2.31It follows that (2.30) are 18 scalar equations for 18 unknowns. By computing the explicit solution of this algebraic system, we find that are given by the equation obtained from (2.27) by replacing {*S*_{n}(*k*),*s*(*k*)} with {*S*_{n}(*k*;*X*_{0}),*s*(*k*;*X*_{0})}. Taking in this equation, we arrive at (2.27). □

### (g) The global relation

The spectral functions *S*(*k*) and *s*(*k*) are not independent but satisfy an important relation. Indeed, it follows from (2.24) that
2.32As , evaluation at (0,*T*) yields the following global relation:
2.33where *c*(*T*,*k*)=*μ*_{3}(0,*T*,*k*).

### (h) The residue conditions

As *μ*_{2} is an entire function, it follows from (2.26) that *M* can only have singularities at the points where the *S*_{n}′*s* have singularities. We infer from the explicit formulae (2.27) that the possible singularities of *M* are as follows:

— [

*M*]_{1}could have poles in*D*_{1}∪*D*_{2}at the zeros of*s*_{33}(*k*),— [

*M*]_{1}could have poles in*D*_{2}at the zeros of (*s*^{T}*S*^{A})_{33}(*k*),— [

*M*]_{2}could have poles in*D*_{1}∪*D*_{2}at the zeros of*s*_{33}(*k*),— [

*M*]_{2}could have poles in*D*_{2}at the zeros of (*s*^{T}*S*^{A})_{33}(*k*),— [

*M*]_{3}could have poles in*D*_{3}at the zeros of (*S*^{T}*s*^{A})_{33}(*k*) and— [

*M*]_{3}could have poles in*D*_{3}∪*D*_{4}at the zeros of*m*_{33}(*s*)(*k*).

We denote the above possible zeros by and assume that they satisfy the following assumption.

### Assumption 2.5

*We assume that*

—

*s*_{33}(*k*)*has n*_{0}*possible simple zeros in D*_{1}*denoted by*—

*s*_{33}(*k*)*has n*_{1}−*n*_{0}*possible simple zeros in D*_{2}*denoted by*— (

*s*^{T}*S*^{A})_{33}(*k*)*has n*_{2}−*n*_{1}*possible simple zeros in D*_{2}*denoted by*— (

*S*^{T}*s*^{A})_{33}(*k*)*has n*_{3}−*n*_{2}*possible simple zeros in D*_{3}*denoted by*—

*m*_{33}(*s*)(*k*)*has n*_{4}−*n*_{3}*possible simple zeros in D*_{3}*denoted by*—

*m*_{33}(*s*)(*k*)*has n*_{5}−*n*_{4}*possible simple zeros in D*_{3}*denoted by**and*—

*m*_{33}(*s*)(*k*)*has**N*−*n*_{5}*possible simple zeros in D*_{4}*denoted by*,

*and that none of these zeros coincide. Moreover, we assume that none of these functions have zeros on the boundaries of the D*_{n}'*s*.

We determine the residue conditions at these zeros in the following:

### Proposition 2.6

*Let* *be the eigenfunctions defined by* (2.14) *and assume that the set* *of singularities are as the above assumption. Then the following residue conditions hold*:
2.34a
2.34b
2.34c
2.34d
2.34e
and
2.34f*where* *and* *θ*_{ij} *is defined by*
2.35*which implies that*

### Proof.

We will prove (2.34a), (2.34c), (2.34e) and (2.34f), the other conditions follow by similar arguments. Equation (2.26) implies the relation
2.36a
2.36b
2.36c
and
2.36dIn view of the expressions for *S*_{1} and *S*_{2} given in (2.27), the three columns of (2.36a) read
2.37a
2.37b
and
2.37cwhile the three columns of (2.36b) read
2.38a
2.38b
and
2.38cthe three columns of (2.36c) read
2.39a
2.39b
and
2.39cand the three columns of (2.36d) read
2.40a
2.40b
and
2.40cWe first suppose that *k*_{j}∈*D*_{1} is a simple zero of *s*_{33}(*k*). Solving (2.37c) for [*μ*_{2}]_{2} and substituting the result into (2.37a), we find
Taking the residue of this equation at *k*_{j}, we find condition (2.34a) in the case when *k*_{j}∈*D*_{1}. Similarly, solving (2.38c) for [*μ*_{2}]_{2} and substituting the result in to (2.38a), we find
Taking the residue of this equation at *k*_{j}, we find condition (2.34c) in the case when *k*_{j}∈*D*_{2}.

In order to prove (2.34e), we solve (2.39a) and (2.39b) for [*μ*_{2}]_{1} and [*μ*_{2}]_{3}, respectively, then substituting the result into (2.39c), we find
Taking the residue of this equation at *k*_{j}, we find condition (2.34e) in the case when *k*_{j}∈*D*_{3}. Similarly, solving (2.40a) and (2.40b) for [*μ*_{2}]_{1} and [*μ*_{2}]_{3}, respectively, then substituting the result into (2.40c), we find
Taking the residue of this equation at *k*_{j}, we find the condition (2.34f) in the case when *k*_{j}∈*D*_{4}. □

## 3. The Riemann–Hilbert problem

The sectionally analytical function *M*(*x*,*t*,*k*) defined in §2 satisfies a Riemann–Hilbert problem which can be formulated in terms of the initial and boundary values of *u*(*x*,*t*). By solving this Riemann–Hilbert problem, the solution of (1.5) (then (1.3)) can be recovered for all values of *x*,*t*.

### Theorem 3.1

*Suppose that u(x,t) is a solution of* (1.5) *in the half-line domain Ω with sufficient smoothness and decays as* . *Then u(x,t) can be reconstructed from the initial value* {*u*_{0}(*x*)} *and boundary values* {*g*_{0}(*t*),*g*_{1}(*t*),*g*_{2}(*t*)} *defined as follows*,
3.1

*Use the initial and boundary data to define the jump matrices J*_{m,n}(*x,t,k*) *as well as the spectral s*(*k*) *and S*(*k*) *by equation* (2.24). *Assume that the possible zeros* *of the functions* *s*_{33}(*k*),(*s*^{T}*S*^{A})_{33}(*k*),(*S*^{T}*s*^{A})_{33}(*k*) *and m*_{33}(*s*)(*k*) *are as in assumption* 2.3.

*Then the solution* {*u(x,t)*} *is given by*
3.2*where M(x,t,k) satisfies the following 3×3 matrix Riemann–Hilbert problem*:

—

*M is sectionally meromorphic on the Riemann k-sphere with jumps across the contours*(*figure*2).—

*Across the contours**M satisfies the jump condition*3.3— .

—

*The residue condition of M is shown in proposition*2.6.

### Proof.

It remains only to prove (3.2) and this equation follows from the large *k* asymptotics of the eigenfunctions, see appendix A. □

### Remark 3.2

According to the coordinate transformation (1.4), we in fact, consider the so-called ‘complex modified KdV-type’ equation in this paper.

## 4. Non-linearizable boundary conditions

A major difficulty of IBV problems is that some of the boundary values are unknown for a well-posed problem. All boundary values are needed for the definition of *S*(*k*), and hence for the formulation of the Riemann–Hilbert problem. Our main result expresses the spectral function *S*(*k*) in terms of the prescribed boundary data and the initial data via the solution of a system of nonlinear integral equations.

### (a) Asymptotics

An analysis of (2.7) shows that the eigenfunctions have the following asymptotics as (see appendix A):
4.1awhere
4.2a
4.2b
4.2cIn the following we just use , , and , so only we compute these functions
4.2dFrom the global relation (2.33) and replacing *T* by *t*, we find
4.3We define functions {*Φ*_{13}(*t*,*k*),*Φ*_{23}(*t*,*k*),*Φ*_{33}(*t*,*k*)} and by:
4.4we can write 13 and 23 entries of the global relation as
4.5aand
4.5bThe functions are analytic and bounded in *D*_{1}∪*D*_{2} away from the possible zeros of *s*_{33}(*k*) and of order *O*(1/*k*) as .

From the asymptotic of *μ*_{j}(*x*,*t*,*k*) in (4.1a), we have
4.6and
4.7a
4.7bwhere
Here, the definition of *Φ*_{j3}(*t*,*k*) can be found in appendix A.

In particular, we find the following expressions for the boundary values:
4.8a
4.8b
and
4.8cWe will also need the asymptotic of *c*_{j}(*t*,*k*),

### Lemma 4.1

*Global relation* (4.5) *implies that the large k behaviour of c*_{j}(*t*,*k*) *satisfies*
4.9

### Proof.

See appendix B. □

### (b) The Dirichlet and Neumann problems

We can now derive effective characterizations of spectral function *S*(*k*) for the Dirichlet (*g*_{0} prescribed), the first Neumann (*g*_{1} prescribed) and the second Neumann (*g*_{2} prescribed) problems.

Define *α* by *α*=*e*^{2πi/3} and let denote the following combinations formed from :
4.10And let *R*(*k*)=*Φ*_{11}(*s*_{13}/*s*_{33})+*Φ*_{12}(*s*_{23}/*s*_{33}).

Let , where and . Similarly, let , where and .

### Theorem 4.2

*Let* . *Let u*_{0}(*x*),*u*≥0, *be a function of Schwartz class*.

*For the Dirichlet problem, it is assumed that the function g*_{0}(*t*),0≤*t*<*T, has sufficient smoothness and is compatible with u*_{0}(*x*) *at x*=*t*=0.

*For the first Neumann problem, it is assumed that the function g*_{1}(*t*),0≤*t*<*T, has sufficient smoothness and is compatible with u*_{0}(*x*) *at x*=*t*=0.

*Similarly, for the second Neumann problem, it is assumed that the function g*_{2}(*t*),0≤*t*<*T, has sufficient smoothness and is compatible with u*_{0}(*x*) *at x*=*t*=0.

*Suppose that s*_{33}(*k*) *has a finite number of simple zeros in D*_{1}.

*Then the spectral function S(k) is given by*
4.11*where*
*and the complex-value functions* *satisfy the following system of integral equations*:
4.12a
4.12b
4.12c*and* *satisfy the following system of integral equations*:
4.13a
4.13b
4.13c
4.14a
4.14b
and
4.14c

(i)

*For the Dirichlet problem, the unknown Neumann boundary values g*_{1}(*t*)*and g*_{2}(*t*)*are given by*4.15a*and*4.15b(ii)

*For the first Neumann problem, the unknown boundary values g*_{0}(*t*)*and g*_{2}(*t*)*are given by*4.16a*and*4.16b(iii)

*For the second Neumann problem, the unknown boundary values g*_{0}(*t*)*and g*_{1}(*t*)*are given by*4.17a*and*4.17b

### Proof.

Representations (4.11) follow from the relation , and system (4.12) is the direct result of the Volteral integral equations of *μ*_{2}(0,*t*,*k*).

(i) In order to derive (4.15a), we note that equation (4.8b) expresses

*g*_{1}in terms of and . Furthermore, equation (4.7) and Cauchy theorem imply and Thus, 4.18Similarly, 4.19where*I*(*t*) is defined by The last step involves using the global relation to compute*I*(*t*) 4.20Using asymptotic (4.9) and Cauchy theorem to compute the first term on the right-hand side of equation (4.20) and using the transformation and in the second term on the right-hand side of (4.20), we find 4.21Equations (4.19) and (4.21) imply This equation together with (4.8b) and (4.18) yields (4.15a).In order to derive (4.15b), we note that (4.8c) expresses

*g*_{2}in terms of , and . Equation (4.15b) follows from expression (4.18) for and the following formulae: 4.22aand 4.22b(ii) In order to derive representations (4.16) relevant to the first Neumann problem, we use (4.8) together with (4.18), (4.21a) and the following formulae: 4.23a 4.23b and 4.23c

(iii) In order to derive representations (4.17) relevant for the second Neumann problem, we use (4.8) together with (4.18) and the following formulae: 4.24aand 4.24b

□

### (c) Effective characterizations

Substituting into system (4.12), the expressions
4.25a
4.25b
4.25c
and
4.25dwhere *ε*>0 is a small parameter, we find that the terms of *O*(1) give and . Moreover, the terms of *O*(*ε*) give and
4.26From equation (4.26), we can get
4.27a
4.27b
and
4.27c

The Dirichlet problem can now be solved perturbatively as follows: assuming for simplicity that *s*_{33}(*k*) has no zeros and expanding (4.15a) and (4.15b), we find
4.28aand
4.28bUsing equation (4.27a) to determine , we can determine *g*_{11}, *g*_{21} from (4.27), then can be found from (4.26). And these arguments can be extended to higher orders and also can be extended to systems (4.13a) and (4.14a), thus yields a constructive scheme for computing *S*(*k*) to all orders.

Similarly, these arguments also can be used to the first Neumann problem and the second Neumann problem. That is to say, in all cases, the system can be solved perturbatively to all orders.

## Funding statement

The work of Xu was partially supported by Excellent Doctor Research Funding Project of Fudan University. The work described in this paper was supported by grants from the National Science Foundation of China (project nos. 10971031; 11271079), Doctoral Programs Foundation of the Ministry of Education of China and the Shanghai Shuguang Tracking Project (project 08GG01).

## Appendix A. The asymptotic behaviour of the functions

We denote some symbols as follows: A1aand A1bwhere and .

From the Lax pair of *μ*
A2Suppose that
A3We substitute equation (A3) into the Lax pair (A2), and compare the order of *k*, we find that
A4a
A4bWe denote the *D*_{l} by .

Then, from *O*(*k*^{3}), we have
A5*O*(*k*^{2}), we get
A6athis implies that
A6b*O*(*k*), we find
A7athis implies that
A7b*O*(1), we have
A8athis implies that
A8b*O*(*k*^{−1}), we get
A9athis implies that
A9b*O*(*k*^{−2}), we get
A10aThis implies that
A10bAlso, from the *x*-part of the Lax pair, we have the following equations:
A11a
A11b
and
A11cThen from the integral contours *γ*_{j}, we can get
A12

## Appendix B. The asymptotic behaviour of *c*_{j}(*t*,*k*)

Let
The global relation shows that
B1and from equation
we get
B2From the second column of equation (B2), we get
B3Suppose
B4where the coefficients *α*_{l}(*t*) and *β*_{l}(*t*), *l*≥0, are independent of *k*. To determine these coefficients, we substitute equation (B4) into equation (B3) and use the initial conditions
Then we get
B5From the first column of equation (B2), we get
B6Suppose
B7where the coefficients *ξ*_{l}(*t*) and *ν*_{l}(*t*), *l*≥0, are independent of *k*. To determine these coefficients, we substitute equation (B7) into equation (B6) and use the initial conditions
Then we get
B8So, from equation (B1) and the asymptotic of *s*_{j3}(*k*) and *s*_{33}(*k*), we get the asymptotic behaviour of *c*_{j}(*t*,*k*) as ,
B9

- Received February 2, 2013.
- Accepted August 1, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.