## Abstract

We consider the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with inhomogeneous Robin boundary data. We study traditionally important problems of the theory of nonlinear partial differential equations, such as the global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behaviour of solutions for large time.

## 1. Introduction

In this paper, we study the initial-boundary-value (IBV) problem for the nonlinear Schrödinger equation posed on the half-line with the inhomogeneous Robin boundary conditions
1.1where Re *β*<0,*λ*=±i.

The nonlinear Schrödinger (NLS) equation is a famous dispersive equation. The remarkable fact is that the NLS equation is integrable in the sense that it admits a multi-soliton solution, satisfies the infinite number of conservation laws and is solvable using an inverse scattering transform. The Cauchy problem for the NLS equation has been studied by many authors (see [1] and references therein). Different types of cubic nonlinearities, including derivatives of the unknown function and the gauge invariant term |*u*|^{2}*u*, were considered previously (see [2–5] and references therein). Hayashi & Naumkin [6], proved that in the case of the Cauchy problem, the time decay rate of the nonlinear term in equation (1.1) is critical with respect to the large-time asymptotic behaviour of solutions. One of the most important developments in this area is the generalization of the Cauchy problem to the case of the IBV problem on the half-line for important nonlinear evolution equations such as NLS equations. The mathematical models lead precisely to problems, where boundary data are non-zero (sometimes called ‘forced problems’). For example, the launching of solitary waves in a shallow water channel, and the excitation of ion-acoustic solutions in a double plasma machine, belong to this class. In ionospheric modification experiments, one directs a radio frequency wave at the ionosphere. At the reflection point of the wave, a sufficient level of an electron-plasma wave is excited to make the nonlinear behaviour important [7]. This may be described by the cubic NLS equation with Dirichlet boundary conditions (*β*=0 in (1.1)). The IBV problem (1.1) also has significant physical implications. For example, it arises from the propagation of optical solitons [8].

For the case of inhomogeneous Dirichlet boundary conditions, there are certain results. Ogawa & Ozawa [9] proved the uniqueness of weak solutions to a mixed problem for the nonlinear Schrödinger equation posed on some uniformly regular domain. Local existence in some Sobolev spaces was obtained by Holmer [10] and Bu [11]. Weder [12] proved that the IBV problem for the forced nonlinear Schrödinger equation with a potential on the half-line is locally and (under stronger conditions) globally well posed. Bu & Strauss [13] proved the existence of a global-in-time solution in the energy space for initial data in *H*^{1} and the boundary data from *C*^{3} with a compact support. Boutet de Monvel *et al*. [14] considered the case of time-periodic boundary data. It was proved that the solution has different asymptotic behaviours in different regions. In Biondini & Bui [15], boundary-value problems for the NLS equations on the half-line with homogeneous Robin boundary conditions were revisited using Bäcklund transformations. The results were illustrated by discussing several exact soliton solutions, which described the soliton reflection at the boundary. There are few results in the case of inhomogeneous Robin boundary conditions. Bu [16] showed that there exists a unique classical solution *u*∈**C**^{0}(**H**^{2})∩**C**^{1}(**L**^{2}) provided that *u*_{0}∈**H**^{2} and boundary data . There are also some results using the inverse scattering techniques (see [17] and references therein). A method for analysing IBV problems, based on ideas of the inverse scattering method, was introduced in Fokas [18]. For a particular class of boundary conditions, called linearizable, the Fokas method can be used to establish large-time asymptotics for integrable equations such as NLS equations (see [19,20]). For the case of homogeneous Robin boundary conditions, the Fokas method yields explicit results (see [17]). As far as we know, there are no results in the case of inhomogeneous Robin boundary conditions. In addition, the advantage of our approach is that it can also be applied to non-integrable equations.

In this paper, we prove the global-in-time existence of solutions of IBV problems with inhomogeneous Robin boundary conditions. In addition, we are interested in studying the asymptotic behaviour of solutions. Hayashi & Kaikina [21] proved that in the case of Robin IBV problems for dissipative equations, the solutions obtain more rapid time decay compared with Cauchy problems. This phenomenon was also observed for some dispersive equations, such as Korteweg–de Vries and intermediate long-wave equations, posed on the positive half-line (see [21,22]). However, as we will show below for the NLS equation with Robin boundary conditions, this is not the case: the cubic nonlinearity |*u*|^{2}*u* is also critical in our problem. Our approach is based on the estimates of the integral equation in Sobolev spaces. To obtain smooth solutions in , we modify a method based on the factorization for the free Schrödinger evolution group [23]. In our case, it should be modified as follows:
where is a Fourier sine transform, operator and To show *a priori* estimates of solutions, we also adopt the operator for the case of the boundary-value problems. The operator was introduced by Ginibre & Velo [24] to study the scattering problem for nonlinear Schrödinger equations, and was used by many authors [1]. However, the operator does not work well in the case of the boundary-value problems owing to different decay properties of the Laplace and Fourier transforms. To avoid this difficulty, we require estimation of the boundary data *u*(0,*t*) more precisely.

To state the results of this paper, we give some notation. We denote 〈*t*〉=1+*t*, {*t*}=*t*/〈*t*〉. The usual direct and inverse Laplace transformations are denoted by and . The Fourier transform and the inverse Fourier transform are defined by
In addition, we introduce a Fourier sine transform denoted by and a Fourier cosine transform denoted by
The usual Lebesgue space , where the norm if and if . The weighted Lebesgue space is , where the norm *k*≥0. The weighted Sobolev space is
where *m*,*k*∈**R**, The usual Sobolev space is **H**^{m}=**H**^{m,0}, so we usually omit the index 0 if it does not cause confusion. In addition, , ∥*φ*∥_{L2}=∥*φ*∥. We denote by the same letter *C* different positive constants.

Now, we state the main results.

### Theorem 1.1

*Let the initial data u*_{0}*∈***H**^{1,1}(**R**^{+}*), boundary data* *and the norm* *be sufficiently small. Then, there exists a unique global solution of the IBV problem (1.1)
**Moreover, there exists* *such that the following asymptotic is valid:
*1.2*for* *where α>0,
*

For the convenience of the reader, we briefly explain our strategy. To estimate the norm ∥*u*∥_{0,1}, we apply the energy method to equation (1.1). To estimate in **L**^{2}, we use the integral equation. To prove the estimate in the norm , we find the solution of (1.1) *u*(*x*,*t*) in the form
Then, from (1.1), we obtain
with Here, the nonlinearity is decomposed into the resonant term −(i/2*t*)|*Ψ*|^{2}*Ψ* and the remainder . The resonant term can be cancelled by the change of the dependent variable using the phase function
In this way, the estimate of *u*(*x*,*t*) follows.

We organize the rest of our paper as follows. In §2, we prove some preliminary estimates for the free Schrödinger evolution group formulated on the half-line. Section 3 is devoted to the proof of theorem 1.1.

## 2. Preliminaries

Denote 2.1and 2.2

We consider the following linear IBV problem on the half-line: 2.3

### Proposition 2.1

*Let the initial data u*_{0}∈**L**^{1} *and boundary data h*(*t*)∈**L**^{1}. *Then, there exists a unique solution u*(*x*,*t*) *of the IBV problem (2.3), which has the integral representation*
2.4*where operators* *and* *are given by* (2.1) *and* (2.2).

### Proof.

Let *K*(*p*)=*λp*^{2}, *K*(*k*(*ξ*))=−*ξ*, Re *k*(*ξ*)>0 for Re *ξ*>0. To derive an integral representation for the solutions of problem (2.3), we suppose that there exists a solution *u*(*x*,*t*) of problem (2.3), which is continued by zero outside of *x*>0. Applying the Laplace transformation with respect to time and space variables to problem (2.3), we find for Re *p*>0 that
2.5Here, the functions and are the Laplace transforms for *u*(*x*,*t*),*u*(0,*t*) and *u*_{x}(0,*t*) with respect to time and space variables, respectively.

As there exists only one root *k*(*ξ*) of equation *K*(*z*)=−*ξ* such that Re *k*(*ξ*)>0 for all Re *ξ*>0, in the expression for the function , the factor 1/(*K*(*z*)+*ξ*) has a pole in the point *z*=*k*(*ξ*). This implies that for solubility of the non-homogeneous problem (2.3), it is necessary and sufficient that the following condition be satisfied:
2.6Therefore, in problem (2.3), we need to fix one boundary data, and the rest of the boundary data can be found from equation (2.6). Thus, for example, if we put the Robin condition , from equation (2.6), we obtain
2.7where function is the Laplace transform of *h*(*t*). Under conditions (2.7) and Re *β*<0, the function is an analytical function in Re *z*>0, Re *ξ*>0. Taking inverse Laplace transforms of (2.5) with respect to time and space variables, we obtain
where operators and are given by and
2.8The function *G*(*x*,*y*,*t*) is defined by the formula
2.9From the Cauchy theorem, *G*(*x*,*y*,*t*)=0 for *x*<0. Now, we consider the case of *x*>0. As e^{ξt}/(*K*(*p*)+*ξ*) has a singularity in the point *ξ*=−*K*(*p*) for all Re *p*=0 by the Cauchy residue theorem and the Jordan lemma, we obtain
Therefore, the first term of formula (2.9) takes the form
2.10As e^{px}((*βp*−1)/(*K*(*p*)+*ξ*)) has singularities in the points *p*=−*k*(*ξ*) (Re *p*<0) and *p*=*k*(*ξ*) (Re *p*>0) by the Cauchy residue theorem and the Jordan lemma, we obtain
Hence,
2.11Thus, from (2.10) and (2.11), we can rewrite the function *G*(*x*,*y*,*t*) in the form for *x*>0,
Changing *p* to i*p*, we have
Thus, integrating by parts, we obtain
Now, we consider the function given by (2.8). In the same way as in the proof of (2.11), taking the residue in the point *p*=−*k*(*ξ*), we obtain, for *x*>0,
Therefore, making the change of variables *k*(*ξ*)=*p*, we obtain
Proposition 2.1 is proved. □

Now, we collect some preliminary estimates of the Green operator given by (2.1). Define

### Lemma 2.2

*The estimates*
2.122.13*are fulfilled if the right-hand sides are bounded*.

### Proof.

By virtue of
2.14we have
Therefore, from the Plancherel theorem, we obtain
2.15In addition, because we obtain
2.16Using (2.15) and (2.16), we have The first estimate of the lemma is proved. Using
we obtain
Therefore, applying via the Plancherel theorem and (2.14), we obtain
2.17The second estimate of the lemma is proved. Now, we obtain the estimate (2.13) of the lemma. Taking we have
2.18where Using , we obtain
2.19From (2.18), we have
where Therefore, applying (2.19) to the above formula, we obtain
2.20where
Using |e^{−λ(y2/4t)}−1|≤*Ct*^{−1/2}*y* and by the Plancherel theorem and (2.12), we have
In addition, because |e^{−λ(y2/4t)}−1|≤*C* and we obtain
Thus, using the Sobolev theorem,
Therefore, from (2.20), the estimate (2.13) follows. The lemma is proved. □

In lemma 2.3, we prove some preliminary estimates of the operator given by (2.2).

### Lemma 2.3

*The estimates*
2.21*are fulfilled if the right-hand sides are bounded. Moreover, the following asymptotics are valid for γ*>0:
2.22

### Proof.

Using , where
2.23from lemma 2.2 by the Plancherel theorem, we have
Therefore, as
2.24we obtain From lemma 2.2, we have
Thus, using
and the estimate (2.24), we prove From lemma 2.2, we have
where *ϕ* is given by (2.23). Thus,
and lemma 2.3 is proved. □

In the proof of estimates of the norm we use the following estimate for *u*(0,*t*).

### Lemma 2.4

*Let u be the solution of the problem (1.1). Suppose that*
*and ε*>0 *is sufficiently small. Then, for t*>1,
2.25

### Proof.

By the Duhamel principle, we find a solution of problem (1.1) in the form
and therefore
Using (2.13) and (2.22),
2.26where
we obtain
2.27Using (2.14), we obtain for *u*_{0}∈**L**^{1},*t*>1. In addition, we have
where
2.28Applying (2.26) to this formula, we obtain
2.29For *ϕ* given by (2.28), we obtain
Therefore, because |*u*|^{2}*u*(*τ*)∈**L**^{1} by the Fubini theorem,
Therefore, , and from (2.29),
2.30where, using lemma 2.2,
By direct calculation,
Therefore, using the Plancherel theorem, we obtain
Applying the above estimate to (2.30), we obtain
2.31Note that , and therefore
Applying the above estimate, (2.27) and (2.31) using a standard continuation argument, we can prove (2.25). The lemma is proved. □

In the proof of estimates of the norm , we use the following estimates.

### Lemma 2.5

*Let* , *then*
2.32*is fulfilled for p*>0, *t*>0, *if the right-hand sides are bounded*.

### Proof.

Taking and and using
we have
Let *Ψ*=*g*(*y*)−e^{−y}*g*_{0}. Applying using (2.14), we obtain
2.33where
2.34
2.35
and
2.36Making the change of variables *p*=*x*/2*t*, we obtain
2.37where
Now, we estimate *R*_{1}, defined in (2.34). As *Ψ*(0,*t*)=0, by direct calculation, we have
2.38Using (2.38) in the same way as in the proof of (2.26), we obtain
and
Therefore, from the Sobolev theorem, we obtain
2.39By analogy, we obtain
2.40where *R*_{2} is given by (2.35). By applying (2.36), (2.37), (2.39) and (2.40) to (2.33), we obtain (2.32). The lemma is proved. □

## 3. Proof of theorem 1.1

Using the Duhamel principle, we find a solution of the problem (1.1) in the form From lemmas 2.2 and 2.3 and the method of Hayashi & Kaikina [21], we obtain the existence of local solutions in the functional space where

### Theorem 3.1

*Let the initial data u*_{0}*∈***H**^{1} *and boundary data* *. Then, for some time T>0, there exists a unique solution u∈***F** *of the boundary-value problem (1.1). If, in addition, we assume that the norm of the initial data* *is sufficiently small, then there exists a unique solution u∈***F** *of (1.1) on a finite time interval [0,T] with T>1/ε*^{2}*, such that the following estimates* *and* *are valid.*

In lemma 3.2, we obtain the optimal time decay estimate of global solutions to the problem (1.1) and the *a priori* estimate of solutions in the norms **Y**.

### Lemma 3.2

*Let the initial data u*_{0}∈**H**^{1}, *boundary data* *and the norm* *be sufficiently small. Then, there exists a unique global solution to the problem (1.1) such that u*∈**C**(**R**^{+};**H**^{1}). *Moreover, the following estimates are valid:*
3.1

### Proof.

Applying the result of theorem 3.1 and using a standard continuation argument, we can find a maximal time *T*>1 such that the inequalities
3.2are true for all *t*∈[0,*T*]. If we prove (3.1) in the whole time interval [0,*T*], then by the contradiction argument, we obtain the desired result of the lemma. In view of the local existence theorem 3.1, it is sufficient to consider the estimates of the solution for time intervals *t*≥1 only.

Let us start with the norm **Y**. To estimate the norm ∥*u*∥_{0,1}, we apply the energy method to equation (1.1). Integrating by parts, we obtain
As *u*_{x}(0,*t*)=(1/*β*)(*h*(*t*)−*u*(0,*t*)), Re *β*<0, we obtain Therefore,
3.3To estimate in **L**^{2}, we use the integral equation
3.4Applying the operator to both sides of equation (3.4), we obtain
3.5Now, we consider the second term of this relation.

As , we obtain
Therefore, using (2.14), and we obtain
where
Using the Plancherel theorem,
and
3.6From lemma 2.4, we have |*u*(0,*t*)|≤〈*t*〉^{−1/2−1/4+γ}*ε*. Applying this estimate in (3.6), we obtain ∥*I*_{3}∥≤*Cε*^{3}.

By virtue of the self-conjugate property of the nonlinear term we have . Thus, from (3.5),
Therefore, from lemmas 2.2 and 2.3,
Thus, by virtue of the Gronwall inequality, we obtain
3.7In the same way, we can prove ∥*u*_{x}∥<*Cε*. So from (3.3) and (3.7), we have the estimate
3.8for all *t*≥0. To prove the estimate in the norm , we find the solution *u*(*x*,*t*) in the form
3.9From equation (1.1), we have
3.10where, from lemma 2.5,
3.11with
3.12for *α*>0. Now, we calculate |*u*|^{2}*u*|_{x=2pt}. Applying (2.13) of lemma 2.2 and (2.22) of lemma 2.3 to (3.9), we obtain
3.13where
and
Using (3.11) and (3.13), from (3.10) finally, we obtain
3.14where
3.15In the same way as in Hayashi & Naumkin [6], we see that the nonlinear term |*Ψ*|^{2}*Ψ* can be removed by the phase function
Indeed, from (3.14), we obtain for new function *v* the following equation Therefore, we obtain
Thus,
3.16Using (3.8) and (3.16) and the contradiction argument, we obtain the result of the lemma. Lemma 3.2 is proved. □

Now, we turn to the proof of the asymptotic formula (1.2) for the solution *u* of the problem (1.1). Using (3.14), we obtain, for the new dependent variable *v*=*BΨ*,
3.17where Integrating (3.17) with respect to *t*, we obtain Making use of the estimate (3.15), we find
3.18for any *t*>*s*>0, where Therefore, there exists a unique final state *v*_{+} such that
3.19for all *t*>0. As |*B*|=1, we write the identity
3.20where
We study the asymptotics in time of the remainder term *Φ*(*t*). We have
Hence, using (3.19), we obtain for any *t*>*s*>0, where Thus, there exists a unique real-valued function such that
3.21for all *t*>0. Representation (3.20) and estimate (3.21) yield
3.22for all *t*>0. Thus, we obtain the large-time asymptotics
with Using (2.26) and (2.22), this completes the proof of asymptotic (1.2). Theorem 1.1 is proved.

## Acknowledgements.

This work was partially supported by CONACYT and PAPIIT IN101311. We thank the unknown referees for useful comments and suggestions.

- Received May 22, 2013.
- Accepted August 23, 2013.

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