In mathematics, a normalized solution to an ordinary or partial differential equation is a solution with prescribed norm, that is, a solution which satisfies a condition like ∫ R N | u ( x ) | 2 d x = 1. {\displaystyle \int _{\mathbb {R} ^{N}}|u(x)|^{2}\,dx=1.} In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.
Definition and variational framework In order to illustrate this concept, consider the following nonlinear Schrödinger equation with prescribed norm:
− Δ u + λ u = f ( u ) , ∫ R N | u | 2 d x = 1 , {\displaystyle -\Delta u+\lambda u=f(u),\quad \int _{\mathbb {R} ^{N}}|u|^{2}\,dx=1,}
where Δ {\displaystyle \Delta } is a Laplacian operator, N ≥ 1 , λ ∈ R {\displaystyle N\geq 1,\lambda \in \mathbb {R} } is a Lagrange multiplier and f {\displaystyle f} is a nonlinearity. If we want to find a normalized solution to the equation, we need to consider the following functional: Let I : H 0 1 ( R N ) → R {\displaystyle I:H_{0}^{1}(\mathbb {R} ^{N})\rightarrow \mathbb {R} } be defined by
I ( u ) = 1 2 ∫ R N | ∇ u | 2 d x − ∫ R N F ( u ) d x {\displaystyle I(u)={\frac {1}{2}}\int _{\mathbb {R} ^{N}}|\nabla u|^{2}dx-\int _{\mathbb {R} ^{N}}F(u)dx}
with the constraint
M = { u ∈ H 0 1 ( R N ) : ∫ R N u 2 = 1 } , {\displaystyle {\mathcal {M}}=\{u\in H_{0}^{1}(\mathbb {R} ^{N}):\int _{\mathbb {R} ^{N}}u^{2}=1\},\ \ \ \ }
where H 0 1 ( R N ) {\displaystyle H_{0}^{1}(\mathbb {R} ^{N})} is the Hilbert space and F ( s ) {\displaystyle F(s)} is the primitive of f ( s ) {\displaystyle f(s)} . A common method of finding normalized solutions is through variational methods, i.e., finding the maxima and minima of the corresponding functional with the prescribed norm. Thus, we can find the weak solution of the equation. Moreover, if it satisfies the constraint, it's a normalized solution.
A simple example on Euclidean space
On a Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} , we define a function f : R 2 → R : {\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} :}
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