发表机构
Savitribai Phule Pune University; Santa Fe Institute; Los Alamos National Laboratory; University of New Hampshire（萨维特里拜·普勒浦那大学; 圣塔菲研究所; 洛斯阿拉莫斯国家实验室; 新罕布什尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文求解了具有标量-标量加矢量-矢量相互作用的1+1维非线性狄拉克方程的精确孤立波解，计算了电荷和能量，并分析了参数空间中束缚态存在条件及峰形转变，最后约化为修正NLSE并讨论稳定性。
AI 中文摘要
我们获得了1+1维非线性狄拉克方程的精确解，形式为ψ(x,t)=e^{-iωt}ψ(x)，对应于标量-标量（SS）加矢量-矢量（VV）相互作用，其相互作用拉格朗日量由L_I = (g^2/(κ+1))[(ψ̄ψ)^{κ+1} + (1/p)(ψ̄γ_μψ ψ̄γ^μψ)^{κ+1}]给出，其中p>0但其他方面任意。我们寻找0<ω<m的解，其中ω和m分别是频率和质量。我们找到了该范围内所有ω值的解。我们计算每个孤立波解的电荷Q和能量E，并探索(p, κ)参数空间中孤立波束缚态存在的区域（即E/Q < m的区域）。我们表明，在所有情况下，虽然E和Q都依赖于耦合常数g，但它们的比值E/Q与g无关。我们进一步发现，在pκ≤1的情况下，所有孤立波的电荷密度只有单峰，而对于pκ>1，存在从双峰到单峰的转变，我们将其确定为ω/m的函数。我们注意到，对于所有p，在κ=2处E/Q作为ω的函数行为存在转变，我们推测这与解在κ=2处不稳定性的开始有关。我们获得了该两参数族的非相对论约化，得到非相对论修正非线性薛定谔方程（NLSE），并讨论了在修正NLSE有效域内单峰孤立波的稳定性。
英文摘要
We obtain exact solutions of the nonlinear Dirac equation in $1+1$ dimensions of the form $ψ(x,t) = e^{-iωt} ψ(x)$ for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by $$L_{I} = \frac{g^2}{κ+1}[(\barψ ψ)^{κ+1} +\frac{1}{p} (\barψ γ_μψ\barψ γ^μ ψ)^{κ+1}]$$ where $p > 0$ but arbitrary otherwise. We look for solutions with $0 < ω< m$ where $ω, m$ are frequency and mass, respectively. We find solutions for all values of $ω$ in this range. We compute the charge $Q$ and the energy $E$ for each of the solitary wave solutions and explore the region in the ($p, κ$) parameter space in which solitary wave bound states exist (i.e., for which $E/Q < m$). We show that for all the cases while both $E$ and $Q$ depend on the coupling constant $g$, their ratio $E/Q$ is independent of $g$. We further find that in case $pκ\le 1$, the charge density for all the solitary waves have only single hump while for $p κ> 1$ there is a transition from double to single hump and we determine it as a function of $ω/m$. We notice that for all $p$ there is a transition at $κ=2$ in the behavior of $E/Q$ as a function of $ω$ which we speculate is related to the onset of instability of the solutions at $κ=2$. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.
Comments21 pages, 13 figures
