Solitary waves in a discrete nonlinear Dirac equation

Abstract

In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross-Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-continuum limit of vanishing coupling). Numerous unexpected features are identified including a staggered solitary pattern emerging from a single site excitation, as well as two-and three-site excitations playing a role analogous to one-and twosite, respectively, excitations of the discrete nonlinear Schrödinger analogue of the model. Stability exchanges between the two-and three-site states are identified, as well as instabilities that appear to be persistent over the coupling strength , for a subcritical value of the propagation constant Λ. Variations of the propagation constant, coupling parameter and nonlinearity exponent are all examined in terms of their existence and stability implications and long dynamical simulations are used to unravel the evolutionary phenomenology of the system (when unstable).

Solitary Waves in the Nonlinear Dirac Equation

“…Firstly, significant steps have been taken in the nonlinear analysis of stability of such models [17,27,127], especially in one-dimensional [27,29,40,128] and two-dimensional settings [49]. Secondly, computational advances have enabled a better understanding of the associated solitary wave solutions and their dynamics [44,47,149,168] also in the presence of external fields [120]. Thirdly, and perhaps most importantly, NLD starts emerging in physical systems which arise in a diverse set of contexts of considerable interest.…”

Integrable semi-discretization of the massive Thirring system in laboratory coordinates

“…The time evolution of U n is obtained by solving the first differential equation of the system (11). Although the representations (13) and (14) express {R n } n∈Z and {Q n } n∈Z in terms of {U n } n∈Z , it may be easier computationally to solve the last two difference equations of the system (11) instantaneously at every time t ∈ R.…”

“…The trivial zero solution satisfies the semi-discrete MTM system (11) and reduces the Lax equations (16) to uncoupled equations for components of ϕ which are readily solvable. Non-trivial solutions of the semi-discrete MTM system (11) will be constructed in future work.…”

“…Without loss of generality, we normalize α = γ = 1 and introduce the parameter h ∈ R by β = −δ = 2i/h. Then, equations (33), (35), and (39) divided by 2i/h give respectively the third, second, and first equations of the system (11). Thus, integrability of the system (11) is verified from the Lax equations (16) with the Lax operators given by (17) and (18).…”

A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

“…This maximum is caused by the fact that, on the one hand, if β falls below a critical value, then α c > β, and the condition of Equation (16) does not hold; hence, the background (flat-state's) instability masks the exponential instability; on the other hand, if β is above that critical value, then β > α c , and the latter instability manifests itself. From a technical standpoint, it is relevant to note in passing that the finite-difference discretization of the first-order spatial and temporal derivatives introduces a number of additional, yet spurious numerical instabilities (associated with complex eigenvalues stemming from the continuous spectrum), disappearing as one approaches the continuum, infinite-domain limit; see [54][55][56] for similar examples with temporal first derivatives in P T -symmetric sine-Gordon and related systems and [98] for such examples with spatial first derivatives in problems featuring Dirac operators. In what we have discussed above, we have not considered these instabilities, focusing on the true dynamical features of the continuum problem.…”