Nonrelativistic asymptotics of solitary waves in the Dirac equation with the Soler-type nonlinearity

Abstract

We use the perturbation theory to build solitary wave solutions ${\varphi }_{\omega }(x)e^{-i\omega t}$ to the nonlinear Dirac equation in ${\mathbb{R}}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\bar{\psi }\psi )\beta \psi$, with $f(\tau )=|\tau {|}^k+o(|\tau {|}^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $\omega \lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schrödinger charge-critical, one has $Q^{\prime }(\omega )<0$ for $\omega \lesssim m$, implying the absence of the degeneracy of zero eigenvalue of the linearization at a solitary wave.

Approximately 44 pages

References in corpus (2)

• Stability of solitary waves and vortices in a 2D nonlinear Dirac model
• Solitary Wave Solutions for the Nonlinear Dirac Equations

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