Cubic NLS and boundary control formulations

PDF p. 1: (1.1)–(1.3); PDF p. 2: (1.5)–(1.6)

Assume countable choice. Take the exterior and truncated obstacle geometry $\mathrm{\Omega }$, ${\mathrm{\Omega }}_0$, $\mathrm{\Gamma }={\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2$ as defined previously. Let $T>0$ and $\lambda \in \mathbb{R}\setminus \{0\}$. For a complex-valued state $u=u(x,t)$, the cubic nonlinear Schrödinger equation on ${\mathrm{\Omega }}_0\times (0,T)$ is

$i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2}u=0.$

Here $i^2=-1$, $u_t=\mathrm{\partial }u\mathrm{/}\mathrm{\partial }t$, and $\mathrm{\Delta }u={\sum }_{j=1}^n{\mathrm{\partial }}_{x_j}^{2}u$; both differential operators act on the real and imaginary parts of $u$. The expression $\mathrm{\mid }u{\mathrm{\mid }}^{2}u$ is the pointwise cubic expression $u\bar{u}u$ whenever these quantities are defined.

A full-boundary Dirichlet control prescribes $u=h$ on $\mathrm{\Gamma }\times (0,T)$; a full-boundary Neumann control prescribes ${\mathrm{\partial }}_{\nu }u=h$ there. The outer-boundary versions instead prescribe $u=h$ on ${\mathrm{\Gamma }}_1\times (0,T)$ and $u=0$ on ${\mathrm{\Gamma }}_2\times (0,T)$, or respectively ${\mathrm{\partial }}_{\nu }u=h$ on ${\mathrm{\Gamma }}_1\times (0,T)$ and ${\mathrm{\partial }}_{\nu }u=0$ on ${\mathrm{\Gamma }}_2\times (0,T)$. The prescribed function $h$ is the boundary control, and a homogeneous boundary condition is one whose prescribed value is zero.

The normal $\nu$ points outward from ${\mathrm{\Omega }}_0$; for a state continuously differentiable up to the boundary, ${\mathrm{\partial }}_{\nu }u={\mathrm{\nabla }}_{x}u\cdot \nu$ means the classical normal derivative, applied componentwise. For less regular states, the equation and boundary equalities are understood only in a separately specified distributional or trace sense for which all terms are defined. No particular weak Neumann trace, solution class, existence or controllability assertion is included in this formulation.

Exterior and truncated obstacle domains

Bounded C1 domains and their outward normals

Classical normal derivative

The Laplacian of a C² function and of a C² vector field

The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i

Real and imaginary parts, complex conjugation, and modulus

Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ

The Axiom of Countable Choice (AC_ω)

The geometry and disjoint boundary pieces are fixed by Exterior and truncated obstacle domains. Its smooth obstacle boundary and the enclosing sphere provide the local smooth boundary pieces of the bounded truncated open set. The outward normal convention is that of Bounded C1 domains and their outward normals, including the countable-choice assumption defined in The Axiom of Countable Choice (AC_ω). In particular the normal on the obstacle boundary is outward from the truncated exterior, not outward from the obstacle.

The classical normal derivative is the gradient dotted with that normal by Classical normal derivative. The displayed spatial Laplacian is the componentwise extension to complex functions of the real Laplacian in The Laplacian of a C² function and of a C² vector field. Once a solution class and the meaning of its boundary values are specified, each listed equation and boundary prescription is a definite condition on $u$ and $h$. The separate weak-interpretation qualification prevents the classical normal derivative from being implicitly applied to arbitrary Sobolev states.

Complex arithmetic and the imaginary unit use The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i; real and imaginary parts, conjugation and modulus use Real and imaginary parts, complex conjugation, and modulus. The time derivative uses Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ, applied to the real two-component map $(\mathrm{Re}u,\mathrm{Im}u)$ in the time coordinate of $(x,t)$.

The frozen original article supplies equations (1.1)–(1.3) and the outer-boundary alternatives (1.5)–(1.6). It is cited for attribution of this article-origin formulation, not as a proof of a prerequisite or of controllability.