Comment on "Non-reciprocal topological solitons in active metamaterials"

We study the problem of information propagation in brain microtubules. After considering the propagation of electromagnetic waves in a fluid of permanent electric dipoles, the problem reduces to the sine-Gordon wave equation in one space and one time dimensions. The problem of propagation of information is thus set.

We study the perturbed sine-Gordon equation $\theta_{tt}-\theta_{xx}+\sin \theta= F(\varepsilon,x)$, where we assume that the perturbation $F$ is analytic in $\varepsilon$ and that its derivatives with respect to $\varepsilon$ satisfy certain bounds at $\varepsilon=0$. We construct implicitly an, adjusted to the perturbation $F$, virtual solitary manifold, which is invariant in the following sense: The initial value problem for the perturbed sine-Gordon equation with an appro...

We consider the sine-Gordon equation in laboratory coordinates in the quarter plane. The first part of the paper considers the construction of solutions via Riemann-Hilbert techniques. In addition to constructing solutions starting from given initial and boundary values, we also construct solutions starting from an independent set of spectral (scattering) data. The second part of the paper establishes asymptotic formulas for the quarter-plane solution $u(x,t)$ as $(x,t) \to \...

We show how the famous soliton solution of the classical sine-Gordon field theory in $(1+1)$-dimensions may be obtained as a particular case of a solution expressed in terms of the Jacobi amplitude, which is the inverse function of the incomplete elliptic integral of the first kind.

One of the difficulties associated with the scattering problems arising in connection with integrable systems is that they are frequently non-self-adjoint, making it difficult to determine where the spectrum lies. In this paper, we consider the problem of locating and counting the discrete eigenvalues associated with the scattering problem for which the sine-Gordon equation is the isospectral flow. In particular, suppose that we take an initially stationary pulse for the sine...

The ratchet dynamics of a kink (topological soliton) of a dissipative sine-Gordon equation in the presence of ac forces with harmonic mixing (at least bi-harmonic) of zero mean is studied. The dependence of the kink mean velocity on system parameters is investigated numerically and the results are compared with a perturbation analysis based on a point particle representation of the soliton. We find that first order perturbative calculations lead to incomplete descriptions, du...

In classical Lorentz-invariant field theories, localized soliton solutions necessarily break translation symmetry. In the corresponding quantum field theories, the position is quantized and, if the theory is not compactified, they have continuous spectra. It has long been appreciated that ordinary perturbation theory is not applicable to continuum states. Here we argue that, as the Hamiltonian and momentum operators commute, the soliton ground state can nonetheless be found i...

An effective description of the inverse spectral data corresponding to the real periodic and quasiperiodic solutions for the sine-gordon equation is obtained. In particular, the explicit formula for the so-called topological charge of the solutions is found and proved. As it was understood already 20 years ago, it is very hard to extract any formula for this quantity from the Theta-functional expressions. A new method was developed by the authors for this goal. In the appendi...

In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-B\"acklund transformation of the supersymmetric sine-Gordon equation. In addition, a detailed description of the associated Darboux transformat...

We use the deformed sine-Gordon models recently presented by Bazeia et al to discuss possible definitions of quasi-integrability. We present one such definition and use it to calculate an infinite number of quasi-conserved quantities through a modification of the usual techniques of integrable field theories. Performing an expansion around the sine-Gordon theory we are able to evaluate the charges and the anomalies of their conservation laws in a perturbative power series in ...