Sine-Gordon solitons in networks: Scattering and transmission at vertices

Abstract

We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws together with successive spacederivatives of sine-Gordon equation. We analytically obtain traveling wave solutions in the form of standard sine-Gordon solitons such as kinks and antikinks for star and tree graphs. We show that for this case the sine-Gordon equation becomes completely integrable just as in case of a simple 1D chain. This simple analysis provides a cornerstone for the numerical solution of the general case, including a quantification of the vertex scattering. Applications of the obtained results to Josephson junction networks and DNA double helix are discussed.

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

“…An extension of the method applied here to networks with the dynamics determined by the Burgers' equation, the Dirac equation with nonlinearity [25], Korteweg-de Vries [26], or the sine-Gordon equation [27] could lead to new insights into bifurcations present in these systems.…”

“…J r,e (x e )(1 − cos(β r,e (x e )))dx e (27) which at this stage is a function of φ 0 . The remaining matching condition e φ e ( e ) = 0 may be reduced to …”

“…Equations (26), (27), and (28) implicitly define the spectral curves k n (N ). While they are simpler than the corresponding exact equations based on Jacobi elliptic functions, they generally remain nonlinear.…”

“…Let us now turn to the three asymptotic regimes R1, R2, and R3 where additional simplifications allow for a more explicit form of the solutions. In the low-intensity regime R1 λ ∼ g|φ e | 2 /k 2 ∝ gI r /k 3 → 0 where κ = k is bounded one may expand oscillatory functions such as sin(k r,e e /2) and consider the leading-order corrections in the continuity condition (26), the total intensity (27), and the quantizations condition (28). In Appendix B, we show how these expansions can be used to find spectral curves k(N ) in the vicinity of the linear spectrum.…”

Drift of Spectrally Stable Shifted States on Star Graphs

“…This constraint was introduced in [25,26] as a condition of the reflectionless transmission of a solitary wave across the vertex of the star graph Γ. The time flow on Γ is given by the following nonlinear Schrödinger (NLS) equation:…”