_{1}

^{*}

In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schr
ödinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following
i
u
_{t} =
u
_{xx} -
g|
u|
^{p}
u,
g ∈
R,
p > 3,
x ∈ Ω = [0,
L], 0 ≤
t < ∞ with initial condition
u (
x,0) =
u
_{0} (
x) ∈
H
^{2} (Ω) and Robin inhomogeneous boundary condition
u
_{x} (0,
t) +
α
u (0,
t) =
R
_{1}(
t),
t ≥ 0 and
u
_{x} (
L,
t) +
α
u (
L,
t) =
R
_{2} (
t),
t ≥ 0 (here
α is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schr
ödinger equation under initial condition and Robin inhomogeneous boundary condition.

This paper is the continuation of an earlier one [

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 x x + k | u | 2 u (1.1)

u ( x , 0 ) = u 0 ( x ) , u ( 0 , t ) = Q ( t ) .

The initial condition u 0 ( x ) , − ∞ < x < ∞ and inhomogeneous boundary condition Q ( t ) , t ≥ 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. [

For the NLS posed in the quarter plane with Robin inhomogeneous condition ( k ∈ R ):

i u t = u x x + k | u | 2 u (1.2)

u ( x , 0 ) = u 0 ( x ) , u x ( 0 , t ) + α u ( 0 , t ) = R (t)

similar results were available [

Solving such problems has important physical and mathematical implications. For example, (1.2) arises in the propagation of the optical solitons [

This paper will investigate a more general version of nonlinear Schrödinger equation i u t = u x x − g | u | p u on a semi-infinite strip x ∈ Ω = [ 0 , L ] , 0 ≤ t < ∞ . Robin type inhomogeneous boundary conditions are imposed on both endpoints. Using PDE method, we will prove the existence of a unique local classical solution.

In this paper, we study the following NLS with initial condition and Robin inhomogeneous boundary condition ( g ∈ R , p > 3 )):

i u t = u x x − g | u | p u , x ∈ Ω = [ 0 , L ] , 0 ≤ t < ∞ (2.1)

u ( x ,0 ) = u 0 ( x ) ∈ H 2 (Ω)

u x ( 0, t ) + α u ( 0, t ) = R 1 (t)

u x ( L , t ) + α u ( L , t ) = R 2 (t)

here α 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 α is an arbitrary real number throughout.

Q 1 ( t ) = u ( 0 , t ) , P 1 ( t ) = u x ( 0 , t )

R 1 ( t ) = P 1 ( t ) + α Q 1 (t)

Q 2 ( t ) = u ( L , t ) , P 2 ( t ) = u x ( L , t )

R 2 ( t ) = P 2 ( t ) + α Q 2 (t)

R 0 = ∑ i = 1 i = 2 sup 0 ≤ t ≤ T ( | R i ( t ) | + | R ′ i ( t ) | )

We assume that u 0 ( x ) ∈ H 2 ( Ω ) , R 1 ( t ) ∈ C 1 ( Ω ) , R 2 ( t ) ∈ C 1 ( Ω ) have appropriate smoothness. In addition, they satisfy the necessary compatibility conditions to ensure the existence of solution at ∂ Ω and t = 0 ,i.e. u x ( 0,0 ) + α u 0 ( 0 ) = R 1 ( 0 ) and u x ( L ,0 ) + α u 0 ( L ) = R 2 ( 0 ) .

Lemma 2.1. Let A = − i D x 2 + i a , D ( A ) = { v : v ∈ L 2 , v x x ∈ L 2 , v ′ ( 0 ) + α v ( 0 ) = v ′ ( L ) + α v ( L ) = 0 } . Then the operator A is the infinitesimal generator of a continuous semigroup of contractors

Proof. Let

Similarly

This shows that

Take the imaginary part of (2.2),

By [

Then (2.3) becomes

If one sets

By Theorem 2.3.3 of [

the following inequality

Theorem 2.2 Local Existence-Uniqueness. For

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

Substituting (2.4) into (2.1) yields

where

One can converts (2.5) to an integral equation:

By similar analysis as in [

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 [

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

The author declares no conflicts of interest regarding the publication of this paper.

Bu, C. (2020) Local Existence and Uniqueness Theorem for a Nonlinear Schrödinger Equation with Robin Inhomogeneous Boundary Condition. Journal of Applied Mathematics and Physics, 8, 464-469. https://doi.org/10.4236/jamp.2020.83036