Sine-Gordon solitons in networks: Scattering and transmission at vertices
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 space-derivatives 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 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.
Introduction. Nonlinear wave dynamics described by the sine-Gordon equation is of importance in a variety of topics in physics, such as as elastic and stress wave propagation in solids, liquids and tectonic plates (see, e.g., Ablowitz and Segur (2006); Rajaraman (1982); Drazin and Johnson (1989); Ablowitz and Clarkson (1991); Kivshar and Agrawal (2003); Dauxois and Peyrard (2006); Cuevas-Maraver et al. (2014)), transport in Josephson junctions Barone and Paternò (1982), and topological quantum fields Rajaraman (1982); Dauxois and Peyrard (2006). Continuous and discrete forms of the sine-Gordon equation have been used so far for the description of wave transport in different media. However, there are structures for which the wave dynamics cannot be described within the traditional continuous or discrete approaches. These are networks and branched structures where the transmission through a branching point (network vertex) should be described by vertex conditions. Early studies of nonlinear evolution equations in branched structures are Christodoulides and Eugenieva (2001); Cuevas and Kevrekidis (2004); Kevrekidis et al. (2003), and in recent few years one can observe rapidly growing interest in nonlinear waves and soliton transport in networks described by nonlinear Schrödinger equation Sobirov et al. (2010); Adami et al. (2011, 2012a, 2012b, 2013); Sabirov et al. (2013). Integrable boundary conditions following from the conservation laws were formulated, and soliton solutions yielding reflectionless transport across the graph vertex were derived in Sobirov et al. (2010), see also Nakamura et al. (2011) for the case of a discrete nonlinear Schrödinger equation. Burioni et al Bur (2006); Burioni et al. (2005) studied the discrete nonlinear Schrödinger equation and computed transport and reflection coefficients as a function of the wavenumber of a Gaussian wave packet and the length of a graph attached to a defect site.
In this paper we address the wave dynamics in networks described by the sine-Gordon equation on metric graphs, which can be used for modeling of soliton transport in DNA double helix, tectonic plates and Josephson junction networks. The latter has attracted much attention in condensed matter physics Giuliano and Sodano (2009, 2013). Another interesting application of sine-Gordon equations, or, more generally, nonlinear Klein-Gordon equations, on metric graphs can be networks of granular chains Kivshar et al. (2003); Kevrekidis et al. (2003). Recently, soliton dynamics in networks was studied by considering the 2D sine-Gordon equation on and junctions Caputo and Dutykh (2014), and the metric graph limit was also studied numerically. See also Uecker et al. (2015) for similar results for the 2D Nonlinear Schrödinger equation on “fat” graphs.
Here we focus on the problem of integrability of sine-Gordon equations on metric graphs and soliton transmission at the graph vertex. In particular, using an approach similar to that of Sobirov et al. (2010), we discuss the conditions under which the sine-Gordon equation is completely integrable and allows exact traveling wave solutions which provide reflectionless transmission of sine-Gordon solitons across vertices. Numerical solutions with scattering at a vertex when these conditions are violated are also presented.
Conservation laws and boundary conditions. For evolution equations on graphs, the connections of the bonds at the vertices are provided by vertex boundary conditions. In case of linear wave equations such conditions follow from self-adjointness of the problem Kostrykin and Schrader (1999); Exner and Kovařík (2015). For nonlinear evolution equations one should use such fundamental laws as energy, flux, momentum and (for sine-Gordon model) topological charge conservations Sobirov et al. (2010); Adami et al. (2011); Caputo and Dutykh (2014). Below we derive such conditions and show the existence of infinitely many conservation laws in our model, which yields the complete integrability of the system.
Most of the 1D sine-Gordon models follow from the Lagrangian density where and are positive constants.
We want to explore a sine-Gordon model on networks modeled by graphs, i.e., system of bonds which are connected at one or more vertices (branching points). The connection rule is called the topology of the graph. When the bonds can be assigned a length, the graph is called a metric graph. The sine-Gordon model on each bond , is given by Lagrangian density
where . In the following we consider a star graph with three semi-infinite bonds connected at the point called vertex of the graph, see Fig. 1. The coordinates are defined as and , where 0 corresponds to the vertex point. The equation of motion derived from the above Lagrangian density leads to the sine-Gordon equation on each bond given as
To formulate physically reasonable vertex boundary conditions (VBC) one can use the continuity of wave function
and fundamental conservation laws such as energy, charge and momentum conservations together with the asymptotic conditions at infinities: and as , and and as , for some integer , .
For the primary star graph in Fig 1, the energy is defined as
where . Then
and by (2) the energy conservation reduces to
For the same star graph as in Fig 1, the charge is given by
From and (2) we obtain the sum rule
The initial boundary problem (IBVP) (1), (2) and (4) with appropriate initial conditions and asymptotic conditions at infinities is now well defined. However, to see the infinite number of constants of motion, we must search for other additional conditions for parameters.
Integrability and traveling wave solutions. We consider the momentum defined by
For , we impose the condition,
which simplifies the sum rule (6) to
Then (8) becomes
Conditions on higher-order space-derivatives may be available from higher-order conservations, where the analysis becomes more laborious. However, they can also be obtained directly from (1), (2) via Thus,
and similarly, taking successive derivatives of (1), we obtain the conditions on higher-order space-derivatives. It should be emphasized that the momentum conservation requires (9)-(11), from which (12) follows. Now we shall prove that equations (2), (11) and (12) give a scaling function which guarantees the infinite number of constants of motion.
Let us introduce two functions defined on the bonds from 1 to as for and for . Thanks to the vertex boundary conditions (VBC) in (11) and (12), together with the continuity condition in (2), both of with and satisfy , where is a solution of the dimensionless sine-Gordon equation
defined on the real line. This fact is identical to the expression of in terms of the function as
of the sine-Gordon equation (13) on the real line also hold on the star graph, because
The conservation of energy , charge and momentum are just special cases.
From now on, we shall prescribe without loss of generality. Eq.(13) has a number of explicit soliton solutions, for instance: kink “+” and anti-kink “-” solutions which can be written as Drazin and Johnson (1989); Ablowitz and Clarkson (1991)
where is the velocity of the kink. Other soliton solutions include breathers, kink-kink collisions and kink-antikink collisions Ablowitz and Clarkson (1991), to name just a few, see also Saadatmand et al. (2015) for further multi-soliton type solutions. If the sum rule in (10) holds, then all these solutions, or, more generally, all solutions of (13), transfer via (14) to solutions on the metric graph. For instance, the kinks then provide reflectionless transmission of energy through the graph vertex, where the speed and energy of a kink traveling in positive direction () split according to the ratios and , respectively. On the other hand, launching two suitably fine tuned kinks on bonds 2 and 3 in negative direction, their joint energy is transmitted to bond 1.
Before entering into the numerical analysis of kink dynamics, we comment on other VBCs originating in the local scattering properties at each vertex. The VBC (2) of continuity and (4) of local flux conservation with are also called VBC. They naturally appear (Caputo and Dutykh (2014), see also Uecker et al. (2015) for a similar construction for the case of the NLS, and (Exner and Kovařík, 2015, Chapter 8) for an overview of related linear results) by considering the 2D sine–Gordon equation on a “fat” graph, i.e., a 2D branched domain with Neumann boundary conditions, where and are the relative widths of the (fat) bonds.
which conserve charge and energy for all values of the . A simple calculation shows both and VBCs can be derived from (10) and (14), but the inverse derivation is not possible. We note that (11) and (18) conserve and , but if conservation of is enforced, then Eq.(18) reduces to (2). Most importantly, (10) and (14) give the existence of the infinite number of constants of motion (as shown in (15),(16)), which is equivalent to the complete integrability of the sine–Gordon equation on the graph.
Vertex transmission. An important issue for wave dynamics in networks is the scattering at vertices. The sum rule in (10) allows the tuning of the vertex scattering to achieve reflectionless transmission. We now give numerical solutions of (1) with , using 2nd order in space and time finite differences, where we first focus on (2) and (4) as VBC, i.e., the case. Figure 2 shows the reflectionless propagation of a kink in the special case that the sum rule (10) holds.
In Fig. 3 we numerically treat the transmission of solitons through the graph vertex when the sum rule (10) is violated. In (a)-(c) we consider the “natural” case , , and the same kink initial condition as in Fig.2. The total energy is still conserved (by (4)), but in contrast to the reflectionless case from Fig.2, there now is significant reflection at the vertex. To demonstrate and quantify the dependence of the vertex transmission on the in some more detail, in Fig. 3(d) we essentially return to the situation of Fig. 2. That is, we set , , but let vary and plot the reflection coefficient , defined as the ratio of the energies in the incoming bond at initial time and at . At , corresponding to Fig. 2, we have , i.e. zero reflection.
Additionally, the red line in Fig. 3(d) shows the analogous simulation for the case of vertex conditions (11) and (18). Again we have zero reflection at , while violating the sum rule gives qualitatively similar but slightly stronger reflections than the case. Two points should be noted: the simulations in Figs. 2 and 3 have also confirmed the conservation up to numerical discretization effects) of and so long as the sum rule (10) holds (see Fig.4); the simulations do not use the soliton properties of the kinks, but only the fact that they are traveling wave solutions for which we have formulas for the initial conditions. Thus, these numerical results can be transfered to general nonlinear Klein-Gordon equations that admit travelling wave solutions.
Other graph topologies. Our results can be extended to other simple topologies such as general star graphs, tree graphs, loop graphs and their combinations.
Exact traveling wave solutions of sine-Gordon models on such graphs with one incoming semi-infinite bond can be obtained similarly to the above case of a star graph with three bonds, leading to generalizations of the sum rule. We illustrate this for the tree graph from Fig. 5, consisting of three “layers” , where run over the given bonds.
On each bond we have a sine-Gordon equation given by (1). Setting for all , the and have to be determined from the sum rule like (10) at each vertex. For instance, at the three nodes in Fig. 5 we need
where at subsequent bonds we also need to take into account the finite propagation length in the previous bonds, for instance
i.e. . Necessarily, the speeds and energies of, e.g., an incoming kink, also split according to rules like (19), such that on each final bond we only have slow and small energy kinks. A similar construction has been done and formalized for the propagation of Nonlinear Schrödinger solitons on tree graphs in Sobirov et al. (2010).
Another graph for which soliton solutions of sine-Gordon models can be obtained is a graph with a loop (see Fig.6), which consists of two semi-infinite bonds connected by bonds having finite lengths . Requiring the conditions
Finally, it can be shown that the above approach can be applied to obtain exact traveling wave solutions of sine-Gordon models on other (than above) graphs consisting of at least two semi-infinite bonds and any subgraph between them. In this case one needs to impose the pertinent vertex conditions like (20) or (22) at the vertices connecting the semi-infinite bonds with the subgraph.
Conclusions. In this work we studied sine-Gordon equations on simple metric graphs, and derived vertex boundary conditions for charge, energy and momentum conservation, and additionally conditions on parameters, for which the problem has explicit analytical soliton solutions. We find the sum rule (10) for bond-dependent coefficients at each vertex of the graph, which makes the sine-Gordon equation on the graph completely integrable. It is shown that the obtained solutions provide the reflectionless transmission of solitons at the graph vertex. This is also illustrated numerically by quantifying the reflections for a case where these conditions are violated, and we discussed how to generalize the results to other graph topologies. The results can be directly applied to several important problems such as Josephson junction network and DNA double helix. In such approach our model corresponds to continuous version of the system considered in Giuliano and Sodano (2009, 2013). Finally, a very important application can be DNA double helix models where the energy transport is described in terms of sine-Gordon equations Yomosa (1983); Yakushevich (2004). Base pairs of the DNA double helix can be considered as a branched system and modeled by a star graph Yakushevich (2004). Then the -bond energy between two base pairs in such system can be characterized by the parameter, .
Acknowledgement. We thank Panayotis Kevrekidis for his valuable comments on this paper. This work is supported by a grant of the Volkswagen Foundation. The work of DM is partially supported by the grant of the Committee for the Coordination Science and Technology Development (Ref.Nr. F3-003).
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