[Paper Review] Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations

Woocheol Choi, Jinmyoung Seok|arXiv (Cornell University)|Jun 2, 2015

Advanced Mathematical Physics Problems14 references5 citations

TL;DR

This paper establishes the existence and radial symmetry of ground state solutions for the pseudo-relativistic nonlinear Schrödinger equation in $\mathbb{R}^n$, and rigorously proves the nonrelativistic limit as the speed of light $c \to \infty$, showing strong convergence in $H^1(\mathbb{R}^n)$ to the ground state solution of the classical nonlinear Schrödinger equation. The results confirm the formal expectation that the relativistic model reduces to the nonrelativistic one in the low-velocity regime.

ABSTRACT

In this paper we study standing waves for pseudo-relativistic nonlinear Schrödinger equations. In the first part we find ground state solutions. We also prove that they have one sign and are radially symmetric. The second part is devoted to take nonrelativistic limit of the ground state solutions in $H^1 (\mathbb{R}^n)$ space.

Motivation & Objective

• To establish the existence of ground state solutions for the pseudo-relativistic nonlinear Schrödinger equation with power-type nonlinearity.
• To prove that these ground states are positive, radially symmetric (up to translation), and have one sign.
• To rigorously justify the nonrelativistic limit of the ground state solutions as the speed of light $c \to \infty$.
• To demonstrate strong convergence of the relativistic ground states to the nonrelativistic ground state in the $H^1(\mathbb{R}^n)$ norm.

Proposed method

• Uses energy minimization on the Nehari manifold to construct ground state solutions in $H^{1/2}(\mathbb{R}^n)$.
• Applies the Taylor expansion to the relativistic kinetic energy operator $\sqrt{-c^2\Delta + m^2c^4} - mc^2$ to derive its nonrelativistic approximation $-\frac{1}{2m}\Delta$.
• Employs weak convergence arguments and compact Sobolev embeddings to analyze the limit of solutions as $c \to \infty$.
• Establishes $L^p$ and $H^1$ boundedness of ground state solutions uniformly in $c \geq 1$ to ensure convergence.
• Uses uniqueness results for the nonrelativistic equation (Kwong's theorem) to identify the limit as the unique nonnegative, radially symmetric ground state.
• Applies a regularity argument to show that very weak solutions of the limit equation are in fact weak solutions in $H^1(\mathbb{R}^n)$.

Experimental results

Research questions

• RQ1 Do ground state solutions exist for the pseudo-relativistic nonlinear Schrödinger equation for all $\mu > 0$ and $p \in (2, 2n/(n-1))$?
• RQ2 Are these ground state solutions positive, radially symmetric, and of one sign?
• RQ3 Does the sequence of ground state solutions $u_c$ converge strongly in $H^1(\mathbb{R}^n)$ as $c \to \infty$?
• RQ4 Does the limit of $u_c$ as $c \to \infty$ coincide with the ground state solution of the classical nonlinear Schrödinger equation?

Key findings

• Ground state solutions exist for all $\mu > 0$, $c \geq 1$, and $p \in (2, 2n/(n-1))$ via minimization on the Nehari manifold.
• All ground state solutions are positive, radially symmetric up to translation, and have one sign, resolving ambiguity in prior works.
• The ground state solutions $u_c$ are uniformly bounded in $L^p(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$ for $c \geq 1$, enabling the limit analysis.
• As $c \to \infty$, the ground state solutions $u_c$ converge strongly in $H^1(\mathbb{R}^n)$ to the unique nonnegative, radially symmetric ground state solution of the classical nonlinear Schrödinger equation.
• The limit solution satisfies the nonrelativistic equation $-\frac{1}{2m}\Delta u + \mu u = |u|^{p-2}u$ and is identified via uniqueness theorems.
• A very weak solution of the nonrelativistic equation is shown to be a weak solution in $H^1(\mathbb{R}^n)$, ensuring regularity of the limit.