Sine-Gordon solitons in networks: Scattering and transmission at vertices
Zarif Sobirov, Doniyor Babajanov, Davron Matrasulov, Katsuhiro Nakamura, Hannes Uecker
SSine-Gordon solitons in networks: Scattering and transmission at vertices
Zarif Sobirov a,c , Doniyor Babajanov b , Davron Matrasulov b , Katsuhiro Nakamura d,e , Hannes Uecker f a Tashkent Financial Institute, 60A, Amir Temur Str., 100000, Tashkent, Uzbekistan b Turin Polytechnic University in Tashkent, 17 Niyazov Str., 100095, Tashkent, Uzbekistan c Faculty of Mathematics, National University of Uzbekistan, Vuzgorodok, Tashkent 100174,Uzbekistan d Faculty of Physics, National University of Uzbekistan, Vuzgorodok, Tashkent 100174,Uzbekistan e Department of Applied Physics, Osaka City University, Osaka 558-8585, Japan f Institut f¨ur Mathematik, Universit¨at Oldenburg, D26111 Oldenburg, Germany
We consider the sine-Gordon equation on metric graphs with simple topologies and derive ver-tex 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 ofstandard sine-Gordon solitons such as kinks and antikinks for star and tree graphs. We show thatfor this case the sine-Gordon equation becomes completely integrable just as in case of a simple 1 D 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 Josephsonjunction networks and DNA double helix are discussed. Introduction.
Nonlinear wave dynamics described bythe sine-Gordon equation is of importance in a varietyof topics in physics, such as as elastic and stress wavepropagation in solids, liquids and tectonic plates (see,e.g., [1–7]), transport in Josephson junctions [8], andtopological quantum fields [2, 6]. Continuous and dis-crete forms of the sine-Gordon equation have been usedso far for the description of wave transport in differentmedia. However, there are structures for which the wavedynamics cannot be described within the traditional con-tinuous or discrete approaches. These are networks andbranched structures where the transmission through abranching point (network vertex) should be described byvertex conditions. Early studies of nonlinear evolutionequations in branched structures are [9–11], and in re-cent few years one can observe rapidly growing interestin nonlinear waves and soliton transport in networks de-scribed by nonlinear Schr¨odinger equation [12–17]. In-tegrable boundary conditions following from the conser-vation laws were formulated, and soliton solutions yield-ing reflectionless transport across the graph vertex werederived in [12], see also [18] for the case of a discretenonlinear Schr¨odinger equation. Burioni et al [19, 20]studied the discrete nonlinear Schr¨odinger equation andcomputed transport and reflection coefficients as a func-tion of the wavenumber of a Gaussian wave packet andthe length of a graph attached to a defect site.In this paper we address the wave dynamics in net-works described by the sine-Gordon equation on metricgraphs, which can be used for modeling of soliton trans-port in DNA double helix, tectonic plates and Josephsonjunction networks. The latter has attracted much atten-tion in condensed matter physics [21, 22]. Another in-teresting application of sine-Gordon equations, or, moregenerally, nonlinear Klein-Gordon equations, on metricgraphs can be networks of granular chains [11, 23]. Re-cently, soliton dynamics in networks was studied by con-sidering the 2D sine-Gordon equation on Y and T junc-tions [24], and the metric graph limit was also studiednumerically. See also [25] for similar results for the 2D Nonlinear Schr¨odinger equation on “fat” graphs.Here we focus on the problem of integrability of sine-Gordon equations on metric graphs and soliton transmis-sion at the graph vertex. In particular, using an approachsimilar to that of [12], we discuss the conditions underwhich the sine-Gordon equation is completely integrableand allows exact traveling wave solutions which providereflectionless transmission of sine-Gordon solitons acrossvertices. Numerical solutions with scattering at a vertexwhen these conditions are violated are also presented. Conservation laws and boundary conditions.
For evolu-tion equations on graphs, the connections of the bonds atthe vertices are provided by vertex boundary conditions.In case of linear wave equations such conditions followfrom self-adjointness of the problem [26, 27]. For nonlin-ear evolution equations one should use such fundamentallaws as energy, flux, momentum and (for sine-Gordonmodel) topological charge conservations [12, 13, 24]. Be-low we derive such conditions and show the existence ofinfinitely many conservation laws in our model, whichyields the complete integrability of the system.Most of the 1D sine-Gordon models follow from theLagrangian density L = (cid:2) (cid:0) u t − a u x (cid:1) − β (1 − cos u ) (cid:3) , where a and β are positive constants. We want to explore O b b b FIG. 1. Sketch of a metric star graph a sine-Gordon model on networks modeled by graphs, i.e.,system of bonds which are connected at one or more ver-tices (branching points). The connection rule is calledthe topology of the graph. When the bonds can be as- a r X i v : . [ n li n . PS ] O c t signed a length, the graph is called a metric graph. Thesine-Gordon model on each bond b k , k = 1 , , , ..., N isgiven by Lagrangian density L k = (cid:20) (cid:0) u kt − a k u kx (cid:1) − β k (1 − cos u k ) (cid:21) , where a k , β k >
0. In the following we consider a stargraph with three semi-infinite bonds connected at thepoint O called vertex of the graph, see Fig. 1. The co-ordinates are defined as x ∈ ( −∞ ,
0] and x , ∈ [0 , ∞ ),where 0 corresponds to the vertex point. The equation ofmotion derived from the above Lagrangian density leadsto the sine-Gordon equation on each bond given as u ktt − a k u kxx + β k sin u k = 0 . (1)To formulate physically reasonable vertex boundary con-ditions (VBC) one can use the continuity of wave function u (0 , t ) = u (0 , t ) = u (0 , t ) (2)and fundamental conservation laws such as en-ergy, charge and momentum conservations to-gether with the asymptotic conditions at infinities: ∂ x u ( x , t ) , ∂ t u ( x , t ) → u ( x , t ) → πn as x → −∞ , and ∂ x u k ( x k , t ) , ∂ t u k ( x k , t ) → u k ( x k , t ) → πn k as x k → ∞ , k = 2 ,
3, for some integer n k , k = 1 , , E ( t ) = (cid:88) k =1 β k (cid:90) B k (cid:20) (cid:0) u kt + a k u kx (cid:1) + β k (1 − cos u k ) (cid:21) dx, (3)where B =( −∞ , , B , =(0 , + ∞ ). Then˙ E = a β u x u t (cid:12)(cid:12)(cid:12)(cid:12) x =0 − a β u x u t (cid:12)(cid:12)(cid:12)(cid:12) x =0 − a β u x u t (cid:12)(cid:12)(cid:12)(cid:12) x =0 , and by (2) the energy conservation reduces to a β u x (cid:12)(cid:12)(cid:12)(cid:12) x =0 = a β u x (cid:12)(cid:12)(cid:12)(cid:12) x =0 + a β u x (cid:12)(cid:12)(cid:12)(cid:12) x =0 . (4)For the same star graph as in Fig 1, the charge Q isgiven by2 πQ = a √ β (cid:90) −∞ u x dx + (cid:88) k =2 a k √ β k + ∞ (cid:90) u kx dx. (5)From ˙ Q = 0 and (2) we obtain the sum rule a √ β = a √ β + a √ β . (6)The initial boundary problem (IBVP) (1), (2) and (4)with appropriate initial conditions and asymptotic condi-tions at infinities is now well defined. However, to see the infinite number of constants of motion, we must searchfor other additional conditions for parameters. Integrability and traveling wave solutions.
We considerthe momentum defined by P = (cid:88) k =1 a k β k (cid:90) B k u kx u kt dx, (7)such that˙ P = a β (cid:20)
12 ( u t + a u x ) − β (1 − cos u ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x =0 − (cid:88) k =2 a k β k (cid:20)
12 ( u kt + a k u kx ) − β k (1 − cos u k ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x k =0 . (8)For ˙ P = 0, we impose the condition, β = β = β = β ( > , (9)which simplifies the sum rule (6) to a = a + a . (10)Then (8) becomes2 β ˙ P = a u x (0 , t ) − a u x (0 , t ) − a u x (0 , t )= − a a a + a ( a u x (0 , t ) − a u x (0 , t )) , where we used (4), (9) and (10) in obtaining the lastexpression. Thus, ˙ P = 0 yields a u x (0 , t ) = a u x (0 , t ),which together with (4), (9) and (10) gives a u x (0 , t ) = a u x (0 , t ) = a u x (0 , t ) . (11)Conditions on higher-order space-derivativesmay be available from higher-order conservations,where the analysis becomes more laborious. How-ever, they can also be obtained directly from (1),(2) via a u xx (cid:12)(cid:12) x =0 = ( u tt + β sin u ) | x =0 =( u ktt + β sin u k ) | x k =0 = a k u kxx (cid:12)(cid:12) x k =0 ( k = 2 , . Thus, a u xx (0 , t ) = a u xx (0 , t ) = a u xx (0 , t ) , (12)and similarly, taking successive x − derivatives of (1), weobtain the conditions on higher-order space-derivatives.It should be emphasized that the momentum conserva-tion requires (9)-(11), from which (12) follows. Now weshall prove that equations (2), (11) and (12) give a scal-ing function which guarantees the infinite number of con-stants of motion.Let us introduce two functions defined on the bondsfrom 1 to k (= 2 ,
3) as v → k ≡ u ( a x √ β , t √ β ) for x < ≡ u k ( a k x √ β , t √ β ) for x ≥
0. Thanks to the vertexboundary conditions (VBC) in (11) and (12), togetherwith the continuity condition in (2), both of v → k with k = 2 , ∈ C ( −∞ , ∞ )) and satisfy v → = v → = v ( x, t ), where v ( x, t ) is a solution of the dimensionlesssine-Gordon equation v tt − v xx + sin v = 0 (13)defined on the real line. This fact is identical to theexpression of u k ( x, t ) in terms of the function v as u k ( x, t ) = v (cid:18) √ βa k x, (cid:112) βt (cid:19) , x ∈ B k ( k = 1 , , . (14)The scaling function v in (14) together with the sumrule (10) guarantees the infinite number of constants ofmotion and thereby the complete integrability of the sine-Gordon equation on the network.In fact, from (14) and the sum rule (10) we find thatall the conservation laws [28, 29] + ∞ (cid:90) −∞ g ( v, ∂ x v, ∂ t v, ∂ x v, ..., ∂ nx ∂ lt v ) dx = const (15)of the sine-Gordon equation (13) on the real line also holdon the star graph, because (cid:88) k =1 (cid:90) B k g (cid:16) u k , a k β − ∂ x u k , β − ∂ t u k , ..., a nk β − n + l ∂ nx ∂ lt u k (cid:17) dx = a ∞ (cid:90) −∞ g (cid:0) v, ∂ x v, ∂ t v, . . . , ∂ nx ∂ lt v (cid:1) dx + ( a + a − a ) + ∞ (cid:90) g (cid:0) v, ∂ x v, ∂ t v, . . . , ∂ nx ∂ lt v (cid:1) dx = const. (16)The conservation of energy E , charge Q and momentum P are just special cases.From now on, we shall prescribe β = 1 without loss ofgenerality. Eq.(13) has a number of explicit soliton solu-tions, for instance: kink “+” and anti-kink “-” solutionswhich can be written as [3, 4] v ( x, t ) = 4 arctan (cid:20) exp (cid:18) ± x − x − νt √ − ν (cid:19)(cid:21) , (17)where | ν | < ν >
0) split according to the ratios a /a and a /a , respectively. On the other hand, launching twosuitably fine tuned kinks on bonds 2 and 3 in negativedirection, their joint energy is transmitted to bond 1. Before entering into the numerical analysis of kink dy-namics, we comment on other VBCs originating in thelocal scattering properties at each vertex. The VBC(2) of continuity and (4) of local flux conservation with β k = 1( k = 1 , ,
3) are also called δ VBC. They naturallyappear ([24], see also [25] for a similar construction for thecase of the NLS, and [27, Chapter 8] for an overview of re-lated linear results) by considering the 2D sine–Gordonequation ∂ t u − ∆ u + sin u = 0 on a “fat” graph, i.e.,a 2D branched domain with Neumann boundary condi-tions, where w /w = a /a and w /w = a /a are therelative widths of the (fat) bonds.Similarly, the so–called δ (cid:48) VBCs [27, Chapter 8] consistof (11) and a u (0 , t ) − a u (0 , t ) − a u (0 , t ) = 0 , (18)which conserve charge and energy for all values of the a k . A simple calculation shows both δ and δ (cid:48) VBCs canbe derived from (10) and (14), but the inverse derivationis not possible. We note that (11) and (18) conserve E and Q , but if conservation of P is enforced, then Eq.(18)reduces to (2). Most importantly, (10) and (14) give theexistence of the infinite number of constants of motion (asshown in (15),(16)), which is equivalent to the completeintegrability of the sine–Gordon equation on the graph. Vertex transmission.
An important issue for wave dy-namics in networks is the scattering at vertices. The sumrule in (10) allows the tuning of the vertex scattering toachieve reflectionless transmission. We now give numer-ical solutions of (1) with β k = 1( k = 1 , , δ case. Figure2 shows the reflectionless propagation of a kink in thespecial case that the sum rule (10) holds.In Fig. 3 we numerically treat the transmission of soli-tons through the graph vertex when the sum rule (10) isviolated. In (a)-(c) we consider the “natural” case a k =1, k = 1 , ,
3, and the same kink initial condition as in Fig.2.The total energy is still conserved (by (4)), but in con-trast to the reflectionless case from Fig.2, there now issignificant reflection at the vertex. To demonstrate andquantify the dependence of the vertex transmission onthe a k in some more detail, in Fig. 3(d) we essentiallyreturn to the situation of Fig. 2. That is, we set a = 1, a = 0 .
7, but let a vary and plot the reflection coefficient R , defined as the ratio of the energies in the incomingbond at initial time t = 0 and at t = 15. At a = 0 . R = 0, i.e. zero reflec-tion.Additionally, the red line in Fig. 3(d) shows the anal-ogous simulation for the case of δ (cid:48) vertex conditions (11)and (18). Again we have zero reflection at a = 0 . δ case. Twopoints should be noted: the simulations in Figs. 2 and3 have also confirmed the conservation up to numericaldiscretization effects) of Q and P so long as the sum rule(10) holds (see Fig.4); the simulations do not use the soli- (a) (b)(c) (d)FIG. 2. (Color online). Numerical solution of (1), (2), (4),with β k = 1, k = 1 , ,
3, and a = 1, a = 0 . a = 0 . x = − ν = 0 .
9, while u , = ∂ t u , ≡ t –dependence of the energies. ton properties of the kinks, but only the fact that they aretraveling wave solutions for which we have formulas forthe initial conditions. Thus, these numerical results canbe transfered to general nonlinear Klein-Gordon equa-tions that admit travelling wave solutions. (a) (b)(c) (d) R δ ‘ VC δ VC FIG. 3. (Color online). (a)-(c) Numerical solution of (1), (2),(4), with β k = 1 for k = 1 , ,
3. (a)-(c) Reflection of incomingkink at the vertex in case that a = a = a = 1, violating(10), initial conditions as in Fig. 2. u is identical to u , andhence u and E in (c) are omitted. (d) (blue line) Depen-dence of the vertex reflection coefficient R = E | t =15 /E | t =0 on a , where a = 1 , a = 0 .
7, hence a = 0 . δ (cid:48) VBC (11) and (18).
Other graph topologies.
Our results can be extendedto other simple topologies such as general star graphs,tree graphs, loop graphs and their combinations. Exact
FIG. 4. (Color online). Time evolution of momenta. Leftpanel: a = 1 , a = 0 . , a = 0 .
3; right panel: a = a = a = 1. traveling wave solutions of sine-Gordon models on suchgraphs with one incoming semi-infinite bond can be ob-tained similarly to the above case of a star graph withthree bonds, leading to generalizations of the sum rule.We illustrate this for the tree graph from Fig. 5, consist-ing of three “layers” b , ( b i ) , ( b ij ), where i, j run overthe given bonds. FIG. 5. A tree graph with three layers, b ∼ ( −∞ , , b , b ∼ (0 , L k ), k = 1 ,
2, and b ij ∼ (0 , + ∞ ) with i, j = 1 , , . . . . On each bond b , b i , b ij we have a sine-Gordon equa-tion given by (1). Setting β = β i = β ij = 1 for all i, j ,the a i and a ij have to be determined from the sum rulelike (10) at each vertex. For instance, at the three nodesin Fig. 5 we needend of b : a = a + a , end of b : a = a + a + a , end of b : a = a + a , (19)and this continues through the layers. By (13) and (14)this is based on scalings such as u ( x, t ) = v ( x/a , t ) and u i ( x, t ) = v ( x/a i , t ) , (20)where at subsequent bonds we also need to take into ac-count the finite propagation length in the previous bonds,for instance u ( x, t ) = v (( x + x ) /a , t ) , x /a = L /a , (21)i.e. x = a L /a . Necessarily, the speeds and ener-gies of, e.g., an incoming kink, also split according torules like (19), such that on each final bond we only haveslow and small energy kinks. A similar construction hasbeen done and formalized for the propagation of Nonlin-ear Schr¨odinger solitons on tree graphs in [12]. FIG. 6. A graph with a loop. b ∼ ( −∞ , , b k ∼ (0 , L k ), k = 1 , . . . , n , where L k = a k L , b n +1 ∼ (0 , ∞ ). 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 con-nected by n bonds having finite lengths L k . Requiringthe conditions a = n (cid:88) k =1 a k = a n +1 (22)for the coefficients, and L k = a k L ( k = 1 , , ...n ) witha constant L , we can write soliton solutions in a similarway as in (20) and (21).Finally, it can be shown that the above approach canbe applied to obtain exact traveling wave solutions ofsine-Gordon models on other (than above) graphs con-sisting of at least two semi-infinite bonds and any sub- graph between them. In this case one needs to imposethe pertinent vertex conditions like (20) or (22) at thevertices connecting the semi-infinite bonds with the sub-graph. Conclusions.
In this work we studied sine-Gordonequations on simple metric graphs, and derived vertexboundary conditions for charge, energy and momentumconservation, and additionally conditions on parameters,for which the problem has explicit analytical soliton solu-tions. We find the sum rule (10) for bond-dependent coef-ficients at each vertex of the graph, which makes the sine-Gordon equation on the graph completely integrable. Itis shown that the obtained solutions provide the reflec-tionless transmission of solitons at the graph vertex. Thisis also illustrated numerically by quantifying the reflec-tions for a case where these conditions are violated, andwe discussed how to generalize the results to other graphtopologies. The results can be directly applied to severalimportant problems such as Josephson junction networkand DNA double helix. In such approach our model cor-responds to continuous version of the system consideredin [21, 22]. Finally, a very important application canbe DNA double helix models where the energy transportis described in terms of sine-Gordon equations [31, 32].Base pairs of the DNA double helix can be consideredas a branched system and modeled by a star graph [32].Then the H -bond energy between two base pairs in suchsystem can be characterized by the parameter, β . Acknowledgement.
We thank Panayotis Kevrekidis forhis valuable comments on this paper. This work is sup-ported by a grant of the Volkswagen Foundation. Thework of DM is partially supported by the grant of theCommittee for the Coordination Science and TechnologyDevelopment (Ref.Nr. F3-003). [1] M. Ablowitz and H. Segur,
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