Review Reports

Round 1

Reviewer 1 Report

The paper needs major clarifications and rewriting. Please see my Review for more details.

Comments for author File: Comments.pdf

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

In the broader and timely context of the experimental, numerical and

theoretical study of various physical systems living on graphs the

authors restricted their attention to the first nontrivial tri-star

graph of Figure 1. They decided to analyze and discuss the dynamics

of non-topological solitons in such a kinematical setting.

The novelty of this study is twofold. Firstly, the motion of the

system in question is assumed controlled by the sine-Gordon (sG)

equation while, secondly, an interesting mathematical concept is

introduced in an attempted reduction of the model into its nonlinear

Schr\"{o}dinger (NLS) analogue.

The methods used are mainly numerical. Typically, the NLS soliton

scattering is treated via an integration of the NLS equation by

means of the conventional Runge-Kutta fourth-order method. The

discretization (using a three-point central difference) is also

routine - in the context of a more ambitious numerical mathematics

one would appreciate seeing at least some comments on the possible

role of the propagation of round-off errors.

On a more general level the authors have shown that the sG -> NLS

transformation can only be well formulated in the arrangement using

the small-amplitude breather solutions in the original setting. In

such a case (characterized, remarkably, up to the approximations

made, by another soliton dynamics) it is pointed out that an ad hoc

modification of the matching conditions at the vertex is necessary.

The results of this paper are also phenomenologically relevant.

Related, in the language of simulations and possible experimental

realizations, to the Josephson junctions. Nevertheless, putting

these terms in the title of the paper can be found slightly

misleading because the essence of the message is , predominantly,

numerical. Illustrated, i.a., by the carefully selected pictures.

Naturally, this can be viewed, after all, just as the acceptable

choice of the strategy of the presentation as made by the authors.

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Dear Authors:

I have reviewed the submitted revised manuscript and am now satisfied by the improvements and corrections that you have made following my suggestions. The paper has been considerably improved and constitutes a new and interesting piece of work in this field.