## Abstract

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is *not* asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period *T* of the boundary data with the square of the length of the interval over *π*.

## 1. Introduction

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems (IBVP) with constant coefficients on the half-line and on finite intervals when the boundary data are periodic. Following the recent investigation of Bona & Fokas (2008), our main concern is the existence of a periodic asymptotic profile for the solutions of such problems. As we shall see, the situation is rather different on the half-line and on finite intervals, because, in the case of a finite interval, the ‘waves’ can be ‘reflected’ from one boundary to the other.

We first look at linear problems on the half-line written in the form
1.1
where *ω* is a polynomial of degree , *q*_{0} is a smooth initial datum, *N*∈{1,…,*n*} is the number of boundary data such that the problem is well posed (Fokas & Sung 1999), and (*f*_{j})_{0≤j≤N−1} are *N* given smooth functions. We assume that *f*_{0},…,*f*_{N−1} are periodic functions of time with the same period and we look for a smooth asymptotic profile *q*_{p} such that

for all

*x*≥0,*t*↦*q*_{p}(*t*,*x*) is a periodic function of time andfor all

*x*≥0, .

In Bona & Fokas (2008), the existence of a periodic profile for the solution of the linear KdV equation on the half-line (corresponding to *ω*(*k*)=−i*k*^{3}) is stated, provided that the initial datum *q*_{0} is homogeneous (*q*_{0}≡0) and that the boundary datum *f*_{0} is periodic (*N*=1 in that case). The proof is based on the new general transform method developed by Fokas (1997; see also Fokas 2002). In this paper, we use Fokas’ method to derive the asymptotic behaviour of the solutions of such equations with periodic boundary data. More precisely, for the solutions of the linear Schrödinger equation, of the linear heat equation and of the linear KdV equation on the half-line, we prove the existence of an asymptotic periodic profile when the boundary data are periodic and we provide explicit formulas for these profiles involving the Fourier coefficients of the boundary data.

We also investigate the asymptotic behaviour of the solutions of linear PDEs with constant coefficients on bounded intervals with periodic boundary data of the form
1.2
where *L*>0 is given, *ω* is a polynomial of degree , *q*_{0} is a smooth initial datum, are the numbers of data at *x*=0 and *L*, respectively, such that the problem is well posed (see Pelloni 2004; Fokas & Pelloni 2005, appendix A), and (*f*_{j})_{0≤j≤N1−1} and (*g*_{j})_{0≤j≤N2−1} are *N*_{1}+*N*_{2} given smooth functions that are compatible with *q*_{0}. We assume that and *g*_{0},…,*g*_{N2−1} are periodic functions of time with the same period and we look for a smooth asymptotic profile *q*_{p} such that

for all

*x*∈[0,*L*],*t*↦*q*_{p}(*t*,*x*) is a periodic function of time andfor all

*x*∈[0,*L*], .

In the case of the linear Schrödinger equation, we use the formula derived in Fokas & Pelloni (2005) to obtain results on the long-time behaviour of the solution. In particular, this formula allows us to give sufficient conditions depending on the link between the length *L* of the interval and the period *T* of the boundary data to obtain periodic solutions, as well as solutions that do *not* have any asymptotic periodic profile.

The outline of the paper is the following. In §2, we consider the linear Schrödinger equation on the half-line (i.e. a problem of the form (1.1) with *ω*(*k*)=i*k*^{2}) with homogeneous initial datum (*q*_{0}≡0) and smooth periodic boundary datum *f*_{0}. Using Fokas’ (2008) method, we prove the existence of a periodic profile *q*_{p} and we provide an explicit formula for that profile involving the Fourier coefficients of *f*_{0} (see theorem 2.5). Section 3 is devoted to another illustration of the efficiency of the integral representation method for the long-time analysis of linear IBVP on the half-line with time-periodic boundary data through two examples: the linear heat equation (*ω*(*k*)=*k*^{2}) and the linearized KdV equation (*ω*(*k*)=−i*k*^{3}). In both cases, we follow the method we used in §2, we prove the existence of an asymptotic profile for the exact solution of the problem and we provide an explicit formula for that profile (see theorems 3.1 and 3.4). In §4, we consider problems of the form (1.2) on a finite interval (0,*L*) with *ω*(*k*)=i*k*^{2}. If *f*_{0} and *g*_{0} are *T*-periodic smooth functions, we show that, if *L*^{2}/*π* and *T* are linearly dependent on , the solution of this linear Schrödinger equation can have, for example, unbounded *L*^{2}-norm () and hence it cannot be asymptotically periodic in general (see theorem 4.4). If *T* and *L*^{2}/*π* are linearly independent on , we show that the solution is not asymptotically periodic in general as well (see theorem 4.9). Finally, we provide numerical experiments in §5 for illustration.

## 2. The linear Schrödinger equation on the half-line

In this section, we consider the linear Schrödinger equation on the half-line with a time-periodic boundary datum and we apply the Fokas method (Fokas 2008) for the representation of its solutions. Using contour deformations of the involved integrals and Cauchy’s residue theorem, we prove the existence of a periodic profile in that case and give an explicit formula for that profile involving the Fourier coefficients of the boundary datum (see formula (2.7) of theorem 2.5). Recall that we provide numerical experiments in §5 for illustration.

### (a) The problem

Following Bona & Fokas (2008), we consider the linear Schrödinger equation on the half-line written in the form of the IBVP,
2.1
This problem is of the form (1.1) with *ω*(*k*)=i*k*^{2}. Note that *N*=1 in that case. Since the problem is linear, we restrict ourselves to the case *q*_{0}≡0. Moreover, we assume that *f*_{0} is a smooth periodic function of time.

### (b) The Fokas method

By applying the method described in Bona & Fokas (2008) to problem (2.1), and defining the following *t*-transform of the boundary datum *f*_{0}
2.2
we obtain the following integral representation formula for the solution *q* of problem (2.1).

### Theorem 2.1

*Assume f*_{0} *is a given smooth periodic function of time. Then the solution q of the corresponding problem (2.1) is given by*
2.3

### Remark 2.2

In case *f*_{0}(0)≠*q*_{0}(0)=0, the above solution *q* of problem (2.1) defined by equation (2.3) averages between *q*_{0}(0)=0 and *f*_{0}(0) in the neighbourhood (0,0). The same remark is true for the representation formula (3.2) of problem (3.1) and for the representation formula (3.5) of problem (3.4).

The remaining part of this section is devoted to the use of the integral representation formula (2.3) for the description of the asymptotic behaviour of *q*.

### (c) Long-time asymptotics

In this section, we look at the asymptotic behaviour of the solution *q* of problem (2.1) when the boundary datum *f*_{0} is a smooth periodic function of time. Our goal is to prove theorem 2.5 that ensures the existence of a periodic profile for *q* and provides an explicit formula for that profile involving the Fourier coefficients of *f*_{0}.

Since equation (2.1) is linear and posed on the half-line, we can assume without loss of generality that the periodic boundary datum *f*_{0} has period 2*π*. Since *f*_{0} is smooth, we write
2.4
where the Fourier coefficients tend to 0 rapidly as |*n*| tends to .

Using formula (2.3), we rewrite the exact solution *q* of problem (2.1) (see lemma 2.3). We first define the six complex paths *γ*_{1}, *γ*_{2}, *γ*_{3}, *Γ*_{1}, *Γ*_{2} and *Γ*_{3} as follows:
2.5

### Lemma 2.3

*Let q be the exact solution of problem (2.1) with 2π-periodic smooth boundary datum f*_{0} *written in the form (2.4). The following identity holds for all t,x*>0:
2.6

### Remark 2.4

Note that, for all *t*,*x*>0, every integrand is a smooth function of *k* since the singularities (in if *n*≤−1 or in if *n*≥1 or in 0 if *n*=0) are removable.

Then, we use complex analysis to derive the asymptotic behaviour of the integrals in formula (2.6).

### Theorem 2.5

*We have the following asymptotic periodic profile for the exact solution of problem (2.1) with smooth 2π-periodic boundary datum f*_{0} *written in the form (2.4): for all x*>0,
2.7

### Remark 2.6

This result gives an explicit description of the asymptotic profile of the exact solution of the linear Schrödinger equation on the half-line (2.1) for homogeneous initial datum and periodic boundary datum.

*Proof*.

For all , the integral in the sum on the right-hand side of formula (2.6) reads
Note that, by contour deformation, we can replace *γ*_{1}+*γ*_{2} by *γ*_{1}+*γ*_{3} in the second above integral to derive that it tends to 0 when *t* tends to uniformly in *n*. Moreover, using Jordan’s lemma, we get the value of the first above integral,

For *n*=0, we first use the linearity of the integral to obtain
Using Jordan’s lemma, we compute directly the first integral in the above right-hand side,
For the second integral, we get by contour deformations and change of variable
Applying Lebesgue’s dominated convergence theorem to the last integral yields
This last integral is equal to i*π* by lemma 2.9. Hence,
and the result is proved. □

### Remark 2.7

Formula (2.7) of theorem 2.5 ensures that the Sobolev regularity (in time) of the asymptotic profile of the exact solution of problem (2.1) is the same as that of the periodic datum *f*_{0} (recall that *f*_{0} is assumed sufficiently smooth anyway).

### Remark 2.8

Further analysis of the convergence of to 0 shows that the convergence in equation (2.7) is uniform in *x* on bounded sets.

The following lemma is useful for the proof of theorem 2.5.

### Lemma 2.9

*The following equality holds*:

*Proof*.

The proof of lemma 2.9 is given in the electronic supplementary material. □

## 3. The linear heat equation and the linear KdV equation on the half-line

This section is devoted to another illustration of the efficiency of Fokas’ method for the analysis of the asymptotic behaviour of the solutions of linear IBVPs on the half-line. As we did in §2, we apply this integral representation method to the solutions of linear evolution partial differential equations with constant coefficients of the form (1.1) on the half-line with time-periodic boundary data and we use contour deformation and Cauchy’s residue theorem to derive the existence of periodic profiles and obtain explicit formulas involving the Fourier coefficients of the boundary data (see theorems 3.1 and 3.4). Since the method is very similar to the one used in the previous section (§2), the results are presented in a more concise form. We first look at the linear heat equation in §3*a*. Then we investigate the case of the linear KdV equation in §3*b*, providing an explicit description of the periodic profile whose existence was stated in Bona & Fokas (2008).

### (a) The linear heat equation on the half-line

In this section, we investigate the long-time behaviour of the solutions of the linear heat equation on the half-line with time-periodic boundary data. This problem is of the form (1.1) with *ω*(*k*)=*k*^{2}. Using the linearity of the problem, we restrict ourselves to the case of homogeneous initial data *q*_{0}≡0. Moreover, we assume that *f*_{0} is a smooth function of *t* and is 2*π*-periodic.

We are interested in the long-time behaviour of the solution of the following problem:
3.1
Applying Fokas’ transformation method (Fokas 2008), we obtain the following representation formula for the exact solution of problem (3.1):
3.2
where *ω*(*k*)=*k*^{2} and is defined in equation (2.2).

Our result on the long-time asymptotics of the solution of problem (3.1) is as follows.

### Theorem 3.1

*We have the following asymptotic periodic profile for the exact solution of problem (3.1) with smooth 2π-periodic boundary datum f*_{0} *written in the form (2.4): for all x*>0,
3.3

*Proof*.

Using the linearity of the equation, we restrict ourselves to boundary data of the form *f*_{0}(*t*)=e^{int}. By a straightforward calculation, we get, for ,
Hence, defining for all *t*,*x*>0,
we get
Using equation (3.2), we derive that for all *t*,*x*>0,
One easily checks that
uniformly in *n* and *x*. Moreover, using Jordan’s lemma, we have

To deal with the case *n*=0, we note that, by contour deformation,
where *Γ* is the complex path defined for by *Γ*(*r*)=*r*−i. Note that
by Jordan’s lemma, and
by lemma 3.3. This completes the proof. □

### Remark 3.2

One observes, in theorem 3.1, the following smoothing effect of the heat equation: for any periodic boundary datum *f*_{0} with (sufficient) Sobolev-type regularity, for all *x*>0, the asymptotic profile of the exact solution of problem (3.1) at point *x* has a Gevrey-type regularity.

The following lemma is useful for the proof of theorem 3.1.

### Lemma 3.3

*Let Γ be the complex path defined for* *by Γ*(*r*)=*r*−i. *One has*

*Proof*.

The proof is similar to that of lemma 2.9. □

### (b) The linearized KdV equation

In this section, we investigate the long-time behaviour of the solutions of the linear Korteweg–de Vries (KdV) equation on the half-line with time-periodic boundary data. This problem is of the form (1.1) with *ω*(*k*)=−i*k*^{3}. Using the linearity of the problem, we restrict ourselves to the case of homogeneous initial datum *q*_{0}≡0. Moreover, we assume that *f*_{0} is a smooth function of *t* and is 2*π*-periodic.

We are interested in the long-time behaviour of the solution of the following problem:
3.4
Applying Fokas’ transformation method, we obtain the following representation formula for the exact solution of problem (3.4):
3.5
where *ω*(*k*)=−i*k*^{3} and is defined in equation (2.2).

We denote by *α* the complex number e^{i 2π/3}. We define the four following complex paths for :

### Theorem 3.4

*We have the following asymptotic expansion for the exact solution of problem (3.4) with smooth 2π-periodic boundary datum f*_{0} *written in the form (2.4): for all x*>0,
3.6

### Remark 3.5

Using the definition of *α*, relation (3.6) also reads
3.7

*Proof*.

Using the linearity of the equation, we restrict ourselves to boundary data of the form *f*_{0}(*t*)=e^{int}. By a straightforward calculation, we get, for ,
Hence, defining for all *t*,*x*>0,
we get
Note that for all *t*,*x*>0, *F*_{1}+*F*_{2} is an analytic function of *k*. Using equation (3.5) and contour deformation, we derive that for all *t*,*x*>0,
One easily checks that
uniformly in *n*. Moreover, using Jordan’s lemma, we have

To deal with the case *n*=0, we note that, by contour deformation,
Using Jordan’s lemma, the first term on the right-hand side of the previous equality is equal to 3. Changing variables and deforming contours in the second term yields
using Lebesgue’s dominated convergence theorem. This last limit is equal to using a calculation similar to that of the proof of lemma 2.9. This completes the proof of theorem 3.4. □

### Remark 3.6

We have seen in §§2 and 3 how we can get asymptotic results for the exact solution of linear evolution partial differential equations on the half-line with time-periodic boundary data. The main difference with the nonlinear integrable equations is that in the latter case we *cannot* eliminate directly the unknown boundary values while applying Fokas’ method. However, these unknowns are characterized via a nonlinear integral equation. The question of extracting useful asymptotic information from this equation is under investigation.

## 4. IBVP over the finite interval

### (a) Introduction

In this section, we consider linear evolution PDEs with constant coefficients on a finite interval (0,*L*) where *L* is a given positive real number. These problems are of the form (1.2). Such problems have been studied in Fokas & Pelloni (2005) by implementing the Fokas transformation method.

We are interested in the long-time behaviour of the solutions of these problems when the boundary data are periodic in time. In particular, just as we did for problems on the half-line in the previous sections (see §§2 and 3), we investigate the existence of a periodic profile for the solutions of such problems. More precisely, if *q*(*t*,*x*) denotes the solution of such a problem for *t*≥0 and *x*∈[0,*L*], we define an asymptotic profile *q*_{p}(*t*,*x*) as a smooth periodic function of *t* satisfying for all *x*∈[0,*L*]
With the linear Schrödinger equation as an example, we present in this section different types of asymptotic behaviours that occur when the boundary data are periodic and share the same period. In contrast to the case of linear problems on the half-line (see theorem 2.5), it turns out that an asymptotic profile does *not* exist in general (see theorems 4.1, 4.4 and 4.9).

### (b) The linear Schrödinger equation

We investigate the long-time behaviour of the solution of the following homogenous linear IBVP:
4.1
where *L* and *T* are given positive real numbers and *f*_{0} and *g*_{0} are given smooth *T*-periodic functions. Note that this problem is of the form (1.2) with *ω*(*k*)=i*k*^{2} and *N*_{1}=*N*_{2}=1. Our analysis is based on the following classic representation formula of the solution of equation (4.1) (see Fokas & Pelloni 2005, equation (3.2)): for all *t*>0 and *x*∈(0,*L*),
4.2
where the function *N* is given by and the functions and are the following *t*-transforms of *f*_{0} and *g*_{0}: for all and *t*≥0,
4.3
Although the exact solution of problem (4.1) is a smooth function of (*t*,*x*), the representation formula (4.2) involves a series whose convergence can be weak (and is usually not uniform on (0,*L*) for example). However, as an exact representation of the solution in terms of transforms of the boundary data, it provides much information in particular about the long-time behaviour.

It turns out that the typical behaviour of the exact solution *q* of equation (4.1) depends on whether *T* and *L*^{2}/*π* are linearly independent on or not. As we shall see, this condition corresponds to the possibility for a periodic signal of frequency 1/*T* given at a boundary to get reflected on the other boundary in a specific way. Section 4*c* is devoted to the case *T* and *L*^{2}/*π* are -linearly dependent. Section 4*d* is devoted to the case *T* and *L*^{2}/*π* are -linearly independent.

In the following, we set *b*=2*π*/*T*. Moreover, for all measurable complex functions *h* defined on (0,*L*), we define
and
where *λ* denotes the Lebesgue measure on . Of course, *L*^{2}(0,*L*) (respectively, ) denotes the space of (classes of) complex functions *h* on (0,*L*) such that (resp. ).

### (c) The dependent case

#### (i) A simple example

Before dealing with the general dependent case, we investigate a simple example. In this section, we assume that *T* and *L*^{2}/*π* are linearly dependent on in the following way:
4.4
Hence, we have *b*=*π*^{2}/*L*^{2}. Moreover, we take as boundary data in problem (4.1) the following functions:
4.5
for all . Our result on the long-time behaviour of *q* in that case is as follows.

#### Theorem 4.1

*Assume the period T of the boundary data f*_{0} *and g*_{0} *defined in equation (4.5) satisfies relation (4.4). Then the exact solution of the linear Schrödinger equation (4.1) on (0,L) satisfies*

#### Remark 4.2

In particular, the solution of problem (4.1) with periodic boundary conditions (4.5) does not become asymptotically periodic.

*Proof*.

Note that formula (4.2) reads
4.6
Using equations (4.3) and (4.5), we derive that
Hence,
4.7
and for all such that |*m*|≥2
4.8
Using lemma 4.3, we get that is the sum of
4.9
and
4.10
Using Hölder’s inequality, we have that the real number (4.10) is non-negative. Moreover, since for all , , we have that the real number (4.9) is equal to . Finally, equation (4.7) implies that . Hence, , and the same property holds for . □

#### Lemma 4.3

*For all sequences of complex numbers* *such that* , *the series* *converges in L*^{2}(*0*,*L*) *and its sum f satisfies*

*Proof*.

Essentially, the proof is based on the identity valid for all ,
where *δ* is the Kronecker’s symbol (i.e. *δ*_{a,b}=1 if *a*=*b* and *δ*_{a,b}=0 otherwise). □

#### (ii) The general dependent case

In this section, we still assume that *T* and *L*^{2}/*π* are linearly dependent on . Since problem (4.1) is linear and homogeneous, one can easily deduce the asymptotic behaviour of the general case (where both *f*_{0} and *g*_{0} are non-zero) from the one where *f*_{0}≡0 or *g*_{0}≡0. Of course, while doing so, one has to keep in mind that the possible explosions as (see theorem 4.1) arising from each boundary datum can interact in a destructive way and indeed lead to a bounded solution. Consider, for example, the case with *b*=*π*^{2}/*L*^{2}=2*π*/*T*. In that case, *N*(*t*,*k*_{±1})=0 and for |*m*|≥2, *N*(*t*,*k*_{m}) is a 2*L*^{2}/*π*-periodic function of *t* (see equation (4.8)). Therefore, *q* is itself a 2*L*^{2}/*π*-periodic function of *t*.

Hence, we can focus without too much loss of generality on the asymptotic behaviour of the solution of (4.1) with *g*_{0}≡0. Any linear relation on between *T* and *L*^{2}/*π* yields a relation of the form
4.11
with relatively primes, *α*≥1, *β*≤−1. Such a couple (*α*,*β*) is uniquely determined. Note that the hypotheses of theorem 4.1 correspond to (*α*,*β*)=(1,−1).

#### Theorem 4.4

*Let T*>0 *denote a period of the smooth function f*_{0}, *and set g*_{0}≡0. *Define b*=2*π*/*T. Denote* *the Fourier coefficients of f*_{0},
4.12
*Assume that T and L*^{2}/*π are linearly dependent on* *so that equation (4.11) holds. Let us denote*
*If there exists* (*n*,*m*)∈*R such that* , *then the solution q of problem* (4.1) *satisfies*
*and hence is not asymptotically periodic. Otherwise, q is a periodic function of period* .

*Proof*.

Formula (4.6) can be rewritten
4.13
Note that for all ,
For all *m*≥1, all and all *t*≥0,
Note that, since *f*_{0} is smooth, the coefficients tend to zero rapidly as . Using equation (4.13), we write
where
and
Note that *q*_{p} is a periodic function and is one of its periods. In particular, ∥*q*_{p}(*t*)∥_{L2(0,L)} is bounded. Moreover, for all *m*≥1, contains at most one term. Using lemma 4.3, we derive that
This proves the result. □

### (d) The independent case

In this section, we consider the long-time behaviour of the solution *q* of problem (4.1) on (0,*L*) with *g*_{0}≡0 and *f*_{0} a *T*-periodic smooth function such that *T* and *L*^{2}/*π* are linearly independent on . Under these assumptions, we shall prove that *q* is *not* asymptotically periodic (see §4*a* for a definition) in general (see theorem 4.9).

In order to prove this result, we recall the following.

### Theorem 4.5 (Kronecker’s approximation theorem).

*For all* , *for all* , *for all ϵ,T*>0 *and all linearly independent real numbers* , *there exists a real number t*>*T and n integers* *such that*

See Apostol (1976, theorem 7.9 and exercise 7 of chapter 7) for a proof of this result.

### Theorem 4.6

*Assume that f, g and h are periodic complex-valued functions defined on* *with h*=*f*+*g. Assume that f and g have positive smallest periods. We denote these periods by T*_{1} *and T*_{2}, *respectively. Assume that T*_{1} *and T*_{2} *are linearly independent on* . *Assume that f is bounded. Then f does not have any open interval of continuity*.

See Olmsted & Townsend (1972, theorem 5) for a proof of this result.

#### (i) Preliminary results

#### Lemma 4.7

*Assume that f, g and h are periodic complex-valued functions. Assume that f and g are continuous functions on* *and that the function*
*tends to* 0 *when t tends to* . *Then h is continuous on* .

*Proof*.

The proof of lemma 4.7 is given in the electronic supplementary material. □

The next theorem states that, if a sum of two continuous periodic functions with uncommensurable smallest positive periods is asymptotically periodic, then it is periodic.

#### Theorem 4.8

*Assume that f, g and h are periodic complex-valued functions defined on* . *Assume that f and g are continuous on* *and that they admit smallest positive periods denoted by T*_{1} *and T*_{2} *respectively. Assume that T*_{1} *and T*_{2} *are linearly independent on* *and that the function*
4.15
*tends to* 0 *when t tends to* . *Then d*≡0. *In other words*,
4.16

*Proof*.

The proof of theorem 4.8 is given in the electronic supplementary material. □

#### (ii) Long-time asymptotics in the independent case

#### Theorem 4.9

*Let us denote by q the smooth solution of problem (4.1) corresponding to a smooth periodic boundary datum f*_{0} *with smallest period T*>0 *and g*_{0}≡0. *Assume that T and L*^{2}/*π are linearly independent on* . *For all x*∈(0,*L*), *the function t*↦*q*(*t*,*x*) *is not asymptotically periodic*.

*Proof*.

For all *t*>0, we have, using equation (4.2),
4.29
in *L*^{2}(0,*L*), where is defined in equation (4.3). Since *f*_{0} is *T*-periodic, we write the Fourier expansion of *f*_{0} in the form (4.12), setting as before *b*=2*π*/*T*. Since *T* and *L*^{2}/*π* are -linearly independent, we have for all
We derive that for all
Hence, using equation (4.29), we can define the functions
and
to have the following decomposition of the function *q*:
Note that *f*_{x} is *T*-periodic and *g*_{x} is 2*L*^{2}/*π*-periodic so that *q* is an almost periodic function (for a definition, see Besicovitch 1932; Bohr 1947). Note that, since *f*_{0} is smooth, *f*_{x} is a smooth function of time. Since *q* is also a smooth function of time, so is *g*_{x}. Assume that for some *x*_{0}∈(0,*L*) there exists a continuous periodic profile *q*_{p}(*t*,*x*_{0}) such that
Since *T* and 2*L*^{2}/*π* are linearly independent on , so are the smallest positive periods of the continuous functions *f*_{x0} and *g*_{x0}. Hence, theorem 4.8 ensures that *d*_{x0}≡0. This yields
Finally, theorem 4.6 ensures that *f*_{x0} does not have any open interval of continuity. This is a contradiction. We derive that *t*↦*q*(*t*,*x*_{0}) is not asymptotically periodic. □

## 5. Numerical experiments

### (a) Linear IBVP on the half-line

#### (i) The Schrödinger equation

We first compute the numerical solution *q* of the homogeneous linear initial boundary value problem (2.1) on the half-line, with the boundary datum . This fits the framework of §2.

For any given positive values of *t* and *x*, the numerical value of *q*(*t*,*x*) is computed by integrating the integrand in relation (2.3) on the line *γ*_{1}+*γ*_{3} (see equations (2.5) and (2.6)). Since this function and all its derivatives have exponentially decaying moduli, we truncate the integral to a bounded interval with an error of at most ϵ/2 where ϵ is a given positive (small) given real number. Then, we use the trapezoidal method to compute an approximation of the integral on the bounded interval with sufficiently many steps to ensure an error of at most ϵ/2; see Flyer & Fokas (2008) for similar numerical calculations. We plot the results obtained for *t*∈[0,20] and *s*∈[0,7] in figure 1*a*.

In order to illustrate the convergence of the solution *q* to the periodic profile provided by theorem 2.5, we plot the real parts of *q* (solid line) and of the profile (dashed line) in figure 1*b* at point *x*=2.0 as functions of time. Of course, the numerical values of *q* are computed as before.

#### (ii) The heat equation

We now compute the numerical solution *q* of the homogeneous linear initial boundary value problem (3.1) on the half-line, with the boundary datum . This fits the framework of the first part of §3. To compute numerical values for *q*, we use contour deformation just as we did for the Schrödinger equation above. We plot the solution we obtain in figure 2*a* for *t*∈[0,20] and *x*∈[0,7]. In order to illustrate the convergence of the solution *q* to the periodic profile provided by theorem 3.1, we plot *q* (solid line) and the profile (dashed line) at point *x*=2.0 as functions of time in figure 2*b* as well.

#### (iii) The linearized KdV equation

As a last illustration for the asymptotics of linear IBVP problems on the half-line with periodic boundary data, we compute the solution *q* of the linear problem (3.4) with the boundary datum
5.1
In figure 3*a*, we plot the numerical results obtained for *t*∈[0,30] and *x*∈[0,4]. In order to illustrate the convergence of the solution *q* to the periodic profile provided by theorem 3.4, we plot *q* (solid line) and the profile (dashed line) in figure 3*b* at point *x*=2.0 as functions of time.

### (b) The linear Schrödinger equation on bounded intervals

#### (i) The dependent case

We investigate the asymptotic behaviour of the solutions of the IBVP (4.1) on the finite interval (0,*L*) with *L*=*π*. We compute the numerical values of the solution *q* using a finite difference method scheme in both time and space, with a CFL number 0.05 and 50 discretization points in space.

Firstly, we consider periodic boundary data *f*_{0} and *g*_{0} defined in equation (4.5) with *b*=1 (i.e. *T*=2*π*). Note that relation (4.4) holds true. We plot in figure 4*a* the numerical values of ∥*q*∥_{L2(0,L)} as a function of time for *t*∈[0,50]. We observe that the *L*^{2}-norm of *q* is equivalent to a linear function of time in that case and hence tends to . This is consistent with theorem 4.1. Note that, with the notations of theorem 4.4, we have *α*=1 and *β*=−1 in that case. Moreover, *f*_{0} is 2*π*-periodic and its only non-zero Fourier coefficients are . Hence, (1,−1)∈*R* and . Therefore, the numerical results of figure 4*a* also illustrate part of theorem 4.4.

Secondly, we consider periodic boundary data *f*_{0} and *g*_{0} defined in equation (4.5) with *b*=2 (i.e. *T*=*π*). Note that relation (4.11) holds, with *α*=1 and *β*=−2. The only non-zero Fourier coefficients of *f*_{0} are (see equation (4.12)). Hence, for all (*n*,*m*)∈*R*, . Theorem 4.4 ensures that *q* is a periodic function of time in that case, with period 2*π*. This is illustrated in figure 4*b* where the numerical values of ∥*q*∥_{L2(0,L)} are plotted as a function of time for *t*∈[0,50] in that case.

As a last illustration in the dependent case, we plot in figure 5*a* the real part of the solution of problem 4.1 with as a function of time (*t*∈[0,50]) and space (*x*∈[0,*π*]). This set of data has been discussed at the beginnig of §4*c*(ii). In particular, one observes that the unbounded behaviours arising from *f*_{0} and *g*_{0} somehow balance each other.

#### (ii) The independent case

We investigate the asymptotic behaviour of the solutions of the IBVP (4.1) on the finite interval (0,*L*) with *L*=1. We compute the numerical values of the solution *q* using a finite difference method scheme in both time and space, with a CFL number 0.05 and 50 discretization points in space.

We consider the boundary data *f*_{0} and *g*_{0} given by equation (4.5) with *b*=1 (i.e *T*=2*π*). Theorem 4.9 ensures that the solution *q* is *not* asymptotically periodic in that case. This is illustrated in figure 5*b* where the *L*^{2}-norm of *q* is plotted as a function of time for *t*∈[0,100]. The global behaviour of the solution is illustrated in figure 6*a* where the real part of the solution is plotted for *t*∈[0,50] and *x*∈[0,1]. Moreover, we plot in figure 6*b* the real part of *q*(*t*,*x*) evaluated at *x*=0.5 as a function of time. Figures 5*b* and 6*a,b* show that, even if the solution has some stability property (the *L*^{2}-norm remains bounded in that case), the two periodic behaviours of the two parts of the solution considered in the proof of theorem 4.9 interact in such a way that no periodic asymptotic behaviour seems to take place. This illustrates the result of theorem 4.9.

## Acknowledgements

The author thanks A.S. Fokas and P.A. Markowich for their ideas and comments on this work. This publication is based on work supported by Award No. KUK-I1-007-43, made by the King Abdullah University of Science and Technology.

## Footnotes

- Received April 11, 2009.
- Accepted July 10, 2009.

- © 2009 The Royal Society