Local Existence and Uniqueness Theorem for a Nonlinear Schrödinger Equation with Robin Inhomogeneous Boundary Condition

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1. Introduction

This paper is the continuation of an earlier one [1] where local existence and uniqueness theorem was presented for a one-dimensional nonlinear Schrödinger equation with initial condition and Dirichlet type inhomogeneous boundary condition on a finite domain $\Omega =\left[\mathrm{0,}L\right]$. As we know, many physically important nonlinear evolution equations in one spatial and one temporal dimensions have been found to possess exact solution by the method of inverse scattering transform (IST). For example, the IST technique has been applied to Cauchy problems in the infinite interval $-\infty <x<\infty $ for decaying [2],periodic [3] and self-similar potentials [4]. When a boundary condition is imposed, sometimes it is called forced problem [5] [6] [7] [8]. The physical importance of forced problems for the nonlinear integrable systems was discussed by Kaup [9]. Existence and uniqueness of the solution to the Korteweg-de Vries equation for $0\le x,t<\infty $ where $u\left(x\mathrm{,0}\right)$ and $u\left(\mathrm{0,}t\right)$ are given have been proven by Bona and Winther [10] [11].

The following nonlinear Schrödinger equation (NLS) posed in the quarter plane with Dirichlet inhomogeneous condition (k is a real constant) has been studied by the author:

$i{u}_{t}={u}_{xx}+k{\left|u\right|}^{2}u$ (1.1)

$u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(0,t\right)=Q\left(t\right).$

The initial condition ${u}_{0}\left(x\right),-\infty <x<\infty $ and inhomogeneous boundary condition $Q\left(t\right)\mathrm{,}t\ge 0$ are imposed. Existence and uniqueness of a global classical solution were proved via PDE method provided that the initial-boundary data are “nice” (cf. [12]). Further, (1.1) is shown to be well-posed [13].

For the NLS posed in the quarter plane with Robin inhomogeneous condition ( $k\in \text{R}$ ):

$i{u}_{t}={u}_{xx}+k{\left|u\right|}^{2}u$ (1.2)

$u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{x}\left(0,t\right)+\alpha u\left(0,t\right)=R(t)$

similar results were available [14] [15].

Solving such problems has important physical and mathematical implications. For example, (1.2) arises in the propagation of the optical solitons [16]. Also, the NLS with an additional term ${u}_{x}$ on the right-hand side and $\alpha \to \infty $ models water waves [17].

This paper will investigate a more general version of nonlinear Schrödinger equation $i{u}_{t}={u}_{xx}-g{\left|u\right|}^{p}u$ on a semi-infinite strip $x\in \Omega =\left[0,L\right],0\le t<\infty $. Robin type inhomogeneous boundary conditions are imposed on both endpoints. Using PDE method, we will prove the existence of a unique local classical solution.

2. Existence and Uniqueness of the Local Solution

In this paper, we study the following NLS with initial condition and Robin inhomogeneous boundary condition ( $g\in \text{R}$, $p>3$ )):

$i{u}_{t}={u}_{xx}-g{\left|u\right|}^{p}u,\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega =\left[0,L\right],0\le t<\infty $ (2.1)

$u\left(x\mathrm{,0}\right)={u}_{0}\left(x\right)\in {H}^{2}(\Omega )$

${u}_{x}\left(\mathrm{0,}t\right)+\alpha u\left(\mathrm{0,}t\right)={R}_{1}(t)$

${u}_{x}\left(L\mathrm{,}t\right)+\alpha u\left(L\mathrm{,}t\right)={R}_{2}(t)$

here $\alpha $ is a real number and $t>0$. Using semigroup technique we prove that there exists a unique classical local solution.

We shall utilize the following notations and assume that $\alpha $ is an arbitrary real number throughout.

${Q}_{1}\left(t\right)=u\left(0,t\right),{P}_{1}\left(t\right)={u}_{x}\left(0,t\right)$

${R}_{1}\left(t\right)={P}_{1}\left(t\right)+\alpha {Q}_{1}(t)$

${Q}_{2}\left(t\right)=u\left(L,t\right),{P}_{2}\left(t\right)={u}_{x}\left(L,t\right)$

${R}_{2}\left(t\right)={P}_{2}\left(t\right)+\alpha {Q}_{2}(t)$

${R}_{0}={\displaystyle {\sum}_{i=1}^{i=2}\underset{0\le t\le T}{\mathrm{sup}}\left(\left|{R}_{i}\left(t\right)\right|+\left|{{R}^{\prime}}_{i}\left(t\right)\right|\right)}$

We assume that ${u}_{0}\left(x\right)\in {H}^{2}\left(\Omega \right)\mathrm{,}{R}_{1}\left(t\right)\in {C}^{1}\left(\Omega \right)\mathrm{,}{R}_{2}\left(t\right)\in {C}^{1}\left(\Omega \right)$ have appropriate smoothness. In addition, they satisfy the necessary compatibility conditions to ensure the existence of solution at $\partial \Omega $ and $t=0$,i.e. ${u}_{x}\left(\mathrm{0,0}\right)+\alpha {u}_{0}\left(0\right)={R}_{1}\left(0\right)$ and ${u}_{x}\left(L\mathrm{,0}\right)+\alpha {u}_{0}\left(L\right)={R}_{2}\left(0\right)$.

Lemma 2.1. Let $A=-i{D}_{x}^{2}+ia$, $D\left(A\right)=\left\{v:v\in {L}^{2},{v}_{xx}\in {L}^{2},{v}^{\prime}\left(0\right)+\alpha v\left(0\right)={v}^{\prime}\left(L\right)+\alpha v\left(L\right)=0\right\}$. Then the operator A is the infinitesimal generator of a continuous semigroup of contractors for. Here a is an appropriate positive constant depending on.

Proof. Let. Then X is a Banach space with a norm equivalent to -norm. Let, then and V are dense in H. From Gagliardo-Nirenberg estimates (cf. [18] for details), we have. Let in H. Then clearly is a Cauchy sequence in X. Completeness of X implies that converges in X and. Also,

Similarly

This shows that. Thus A is closed. To show that the resolvent set of A contains, let. Consider

(2.2)

Take the imaginary part of (2.2),

(2.3)

By [18],there exists such that

Then (2.3) becomes

If one sets then

By Theorem 2.3.3 of [19],for the operator maps 1-1 onto H. Now let. For, by taking the real part of (2.2), one has

the following inequality thus. By Hille-Yosida Theorem (cf. [20]), the unbounded and linear operator A is the infinitesimal generator of a continuous semigroup of a contractions for.

Theorem 2.2 Local Existence-Uniqueness. For, , there exists a unique classical solution u for Equation (2.1) such that with either as or.

To prove this theorem, we first apply the following transformation:

(2.4)

Substituting (2.4) into (2.1) yields

(2.5)

where

One can converts (2.5) to an integral equation:

By similar analysis as in [12],G is locally Lipschitz in v under the norm of uniformly on and for each, G is continuous from into (note for any T). Thus one can use Theorem 6.1.7 in [20] to obtain the local existence-uniqueness theorem for (2.5) with with either as or. Since and uniformly on for any, we conclude immediately that u satisfies (2.1) and our existence and uniqueness are proved.

3. Conclusion

Famous nonlinear partial differential equations like nonlinear Schrödinger equation have important applications when the boundary value is not zero. For such equation posed in a semi-infinite strip, we used PDE method to prove that there exists a unique classical local solution, via semigroup theory. The PDE method presented in this paper to study the NLS is an approach different from the IST method in [7]. Along with [12],we try to provide some answers to the questions raised in [9]. There are some further research that could be done in this area. For example, we believe that well-posedness of (2.1) is a subject of research. Also, the existence of a global solution is an open problem. We will continue our work and report any further advances.

Acknowledgements

This research was supported by the William R. Kenan Jr. Professorship, a Brachman Hoffman Small Grant and a Wellesley College Faculty Award.

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