Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

Journal Article · 22 January 2020 · Journal of Physics. A, Mathematical and Theoretical

DOI: https://doi.org/10.1088/1751-8121/ab60e7 · OSTI ID:1798118

Cooper, Fred  ^[1]; Khare, Avinash ^[2]; Quintero, Niurka R.  ^[3]; Sánchez-Rey, Bernardo ^[3]; Mertens, Franz Georg  ^[4]; Saxena, Avadh Behari  ^[5]

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1. Santa Fe Inst. (SFI), Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2. Savitribai Phule Pune University (India)
3. Universidad de Sevilla (Spain)
4. Univ. of Bayreuth (Germany)
5. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Here, the damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter $$\kappa$$ is analyzed, when the external force is periodic in space and given by $$\mathcal{f(x) = r}$$ cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K , and the soliton width $$l_s$$, which is a function of the nonlinearity $$\kappa$$ as well as the initial frequency ω₀ of the solitary wave, there is a transition (at fixed ω₀) from trapped to unbound behavior of the soliton, which depends on the parameters $$\mathcal{r}$$ and K of the external force and the nonlinearity parameter $$\kappa$$. We previously studied this phenomena when $$\kappa$$ = 1 where we showed that for λ $$\gg l_s$$ the soliton oscillates in an effective potential, while forλ $$\ll l_s$$ it moves uniformly as a free particle. In this paper we focus on the $$\kappa$$ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of $$\mathcal{r}$$ and K when $$\kappa$$ = 1/2,1,2 at fixed value of the frequency ω₀. Since the solitary wave gets narrower for fixed ω₀ as a function of $$\kappa$$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $$\kappa$$.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: Alexander von Humboldt Foundation; Indian National Science Academy (INSA); Ministerio de Ciencia, Innovacion y Universidades of Spain; Ministerio de Economía y Competitividad (MINECO); USDOE Laboratory Directed Research and Development (LDRD) Program
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1798118
• Report Number(s): LA-UR--19-29780
• Journal Information: Journal of Physics. A, Mathematical and Theoretical, Journal Name: Journal of Physics. A, Mathematical and Theoretical Journal Issue: 7 Vol. 53; ISSN 1751-8113
• Publisher: IOP PublishingCopyright Statement
• Country of Publication: United States
• Language: English