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44
A solution of nonlinear Schrodinger equation
on metric graphs
K. K. Sabirov1, A. R. Khalmukhamedov2
1National University of Uzbekistan, 4 University St., 100074, Tashkent, Uzbekistan 2Tashkent branch of Moscow State University named after L. V. Lomonosov, 22 Amir Temur St., 100060, Tashkent, Uzbekistan
PACS 02.30.Jr,02.30.Ik DOI 10.17586/2220-8054-2015-6-2-162-172
We treat the Nonlinear Schrodinger equation (NLSE) on Metric graph. An approach developed earlier for NLSE on interval [14], is extended for star graph. Dirichlet boundary conditions are imposed at the ends of bonds are imposed, while continuity conditions are chosen at the vertex of graph.
Keywords: metric graphs, nonlinear Schrodinger equation, solitons.
Received: 2 February 2015
1. Introduction
Wave transport in the nonlinear regime, described by nonlinear evolution equations, such as the nonlinear Schrodinger, Korteweg de Vries and sine Gordon equations, has attracted much attention in different areas of physics over the past five decades (see, e.g. books [1] - [5]).
Recently, one can observe a growing interest in particle and wave transport in branched, network type structures [6] - [13]. Such problem is of importance for different topics in physics, such as hydrodynamics, acoustics, optics, cold atom physics and condensed matter physics. Soliton solutions and connection formulae were derived for simplest graphs in the Ref. [6]. The problem of fast solitons on star graphs was treated in the Ref. [7], where the estimates for the transmission and reflection coefficients were obtained in the limit of very high velocities. The problem of soliton transmission and reflection was studied in [9] by numerically solving the stationary NLSE on graphs. The dispersion relations for linear and nonlinear Schrodinger equations on graphs were discussed in [10]. Ref. [11] treated the stationary NLSE in the context of scattering from nonlinear networks. The stationary NLSE with power focusing nonlinearity on star graphs was studied in recent papers [7,8], where the existence of nonlinear stationary states were shown for 8—type boundary conditions. In [13], the exact analytical solutions of the stationary NLSE for simplest graphs were obtained.
In this work, we treat time dependent NLSE on metric graphs by considering the simplest topology, a star graph. Unlike the case of the NLSE on an interval, in the case of the graph, the NLSE becomes a multicomponent equation, with the components related to each other through the boundary conditions given at the graph vertex.
Our aim is to solve the (cubic) nonlinear Schroodinger equation on metric graphs. The latter are systems consisting of bonds which are connected at the vertices [16] to a
rule which is called the topology of a graph. The topology of a graph can be relayed in terms of a so-called adjacency matrix, which can be written as [17]:
„ _ „ _ J 1, if i and j are connected; . .
Cij = Cji = \ 0, otherwise, i,j = 1,2''', V
In the following, we consider the so-called primary star graph, consisting of three bonds connected at single vertex. However, our results can be extended to any arbitrary topology. Our approach is based on extending the method proposed by Fokas and its solution for the NLSE on a finite interval [14]. In Refs. [14, 15], an effective method allowing one to obtain a general solution for the NLSE on a finite interval [14] and on a half-line [15] was developed. As it will be shown below, this method can be adapted to the case of the NLSE on a metric graph. Thus, the problem we are going to solve is the nonlinear Schrodinger equation on a graph with bonds bj ~ (0,Lj), j = 1,2,3, which can be written as:
d d2
+ dx2 qj- 2A|gj |2qj = 0, (1)
A = ±1, j = 1, 2, 3, Li <xi < 0, 0 <x2,3 < L2>3, 0 < t < T.
The initial conditions are given as:
qj(xj, 0) = qoj(xj), j = 1, 2, 3, Li < xi < 0, 0 < X2,3 < ¿2,3, (2)
The following boundary conditions provide matching of the bonds at the vertex:
qi(0,t) = q2(0,t) = q3(0,t) = go(t), 0 < t < T (3)
qj(Lj,t) = foj(t), 0 < t < T, j = 1, 2, 3, (4)
5 5 5
—qi(0,t) = —q2(0,t) + —q3(0,t), 0 <t<T (5)
Furthermore, we define the following functions:
d d gij(t) = d^qj(0,t), gij(t) = ~dx-qj (Lj,t).
These functions are considered to be unknown and will be found subsequently.
The difference between Eq. (1) and with that treated in Ref. [14] is caused by bond indices, j. In other words, Eq. (1) is a multicomponent equation in which each component is related to others through the boundary conditions (3) - (5). As we will see below, this makes it possible to rewrite most of the results derived in [14] for the case of metric graphs.
2. Description of the approach
The method we are going to utilize includes three steps [14]. The first step consists of Riemann-Hilbert (RH) problem formulation under the assumption of existence. Following Ref. [14], we assume that there exists a smooth solution q(x,t) = {qi(xi,t), q2(x2,t),
q3(x3,t)}.
Applying the spectral analysis to the Lax pair, we write q(x, t) = {qi(xi,t), q2(x2,t), q3(x3,t)} in terms of the solution of a 2 x 2-matrix RH problem defined in the complex k-plane [14]. Such a problem is uniquely defined in terms of the spectral functions which are given as:
{«j (k),bj (k)}, {Aj (k),Bj (k)}, {Aj (k), Bj (k)}. (6)
These functions are defined in terms of the functions:
qoj (Xj), {go(i),gij (t)}, / (t)/ (t)}, (7)
respectively. Here, the functions g0(t, g1j-(t) and /1j-(t) denote the unknown boundary values for the solution to the NLSE and its derivatives (see Eqs. (3)).
Following Ref. [14], one can show that the spectral functions (6) are not independent, but they satisfy the global relation:
(a;aj + Abje2ifcLjbj)Bj - (bjaj + dje2ikLjbj)Aj = e4ifc2Tc+(k), k G cj, (8)
where c+(k) has the same meaning as in [14].
The second step implies proof of the existence of the solution to the NLSE, assuming that the above spectral functions satisfy the global relation. The spectral functions given in (6) can be written in terms of the (smooth) functions (7). We also define q(x,t) = {q1(x1 ,t),q2(x2, t), q3(x3, t)} in terms of the solution of the RH problem formulated in Step 1. Assuming that smooth functions g1j- (t) and /1j- (t) exist such that the spectral functions (6) satisfy the global relation (8), one can prove that:
(i) q(x, t) = {q1(x1, t), q2(x2, t), q3(x3, t)} is defined globally for all L1 < x1 < 0, 0 < x2,3 < L2,3, 0 < t < T.
(ii) q(x,t) = {q1(x1, t), q2(x2, t), q3(x3, t)} solves the NLSE.
(iii) q(x,t) = {q1(x1,t),q2(x2,t),q3(x3,t)} satisfies the given initial and boundary conditions:
qj, 0) = qoj(x) qj(0, t) = £o(t) qj(Lj, t) = /oj(t). A byproduct of this proof is that:
5 5
qj(0, t) = g1j(t) and — qj (Lj, t) = f1j(t).
dxj SX;
Finally, the third step presents an analysis of the global relation treated in the second step. Namely, for given qoj, go, /oj-, one can show that the global relation (8) characterizes g1j and /1j through the solution of a system of nonlinear Volterra integral equations.
Furthermore, following the Ref. [14], we introduce the eigenfunctions,
< ^jra)(x, t, k) I , such that: I j J n=1
^j1)(0,T,k) = I, ^j2)(0, 0,k) = I, ^j3)(Lj, 0,k) = I, j (Lj ,T,k) = I, j = 1, 2, 3, (9)
j
can be written in terms of the matrices
s j, S j, SL as:
with being the 2 x 2 matrices, I = diag(1,1). One can show that these eigenfunctions
jjj
1
sj(k) = ^j3)(0, 0, k), Sj(k) = (e2ifc2TCT3^j2)(0,T,k)e-2№)"
SL(k) = (e2ifc2 ^(3) (Lj ,T,k)e-2ifc2TCT3 )-1, (10)
where a3 = diag(1, -1).
3. Lax pair and its solutions
The Lax pair for our problem can be written as [20]:
d d
—^j + ik(j3,j = Qj , —^j + 2ik2ô"3,j ^j = Q j ^j, (11)
where ^j(x,t,k) is a 2 x 2 matrix-valued function, â3,j is defined by:
¿3,j■ = [^3,j, ■] ^3,j = diag(1, -1), (12)
and the 2 x 2 matrices Qj, Qj are given as:
Q ( t) = ( 0 qj (xj ,t) \ Qj (Xj ^=1 Aqj (xj ,t) 0 J
d
(Qj(xj,t, k) = 2kQj - «d^Qja3,j - |2^3,j, A = ±1. (13)
Furthermore, we assume that there exists a sufficiently smooth solution qj (xj, t), j = 1, 2, 3, xi G [Li, 0], X2,3 G [0,^2,3], t G [0,T], of NLSE. A solution of equation (11) is given by [14]:
^(*)(xj ,t,k) = I + i e-i(kxj+2fc2i)-3,j Wj (y,r,k), (14)
(x j * )
where the closed 1-form Wj is defined by:
Wj = ei(fcx+2fc2 (Qj ^ dx + Qj ^ dt), (15)
(xj*,t*) is an arbitrary point in the domain xi e [Li,0], x2,3 e [0,L2,3], t e [0,T], and the integral denotes a line integral connecting smoothly the points indicated.
Following Ref. [14], it can be shown that the functions are related by these equations:
^.3)(xj,t, k) = 42)(xj,t, k)e-i(kxj+2fc2i)-3,jSj(k), (16)
^j.i)(xj,t, k) = ^j2)(xj,t, k)e-i(kxj+2fc2i)-3,j Sj(k), (17)
^54)(xj,t,k) = j(xj, t, k)e-i(kxj+2fc2SL(k), (18)
and one can find from Eq. (16) at xj = t = 0, s(k) = ^j3)(0,0,k). Finally, from Eqs. (17) j = L j,
-i
and (18) at xj = Lj, t = T we have:
SL(k) = (e2ifc2T<^ ^j3)(Lj ,T,k)) ' and
^j4)(xj, t, k) = ^j2)(xj, t, k)e-i(kxj+2fc2i)-3,j (s(k)eifcLjSL(k)) . (19)
4. The global relation
As was mentioned before, the spectral functions a(k), bj(k), Aj(k), Bj(k), Aj(k), Bj(k) are not independent, but they satisfy the global relation (8), where c+(k) denotes the i'Lj
element of — [exp(iky<r3,j)](Qj)(y,T, k)dy, and j is defined by an equation similar
to ^j) with I replaced by — I . The proof is the same as in the case of the NLSE for
j o o
the finite interval treated in [14]. We now introduce Mj (xj, t, k), defined by:
r-T
j =
ß_ ..(4)(1)
i ßj
aj (k)
J
M.
„(W2)
ßj_,^(3)(2)
dj (k)
M
(+)
ß
(3)(3) ßj
(1)(3)'
dj (fc)
M
(-)
ß
(4)(4) ßj
(2)(4)'
aj (fc)
, arg k G , arg k G arg k G arg k G
»4
n L2
3n
n,
3n
y2n
(20)
where the scalars dj (k) and aj (k) are defined below. These definitions imply:
det Mj (xj ,t,k) = 1
(21)
and
Mj (xj ,t,k)
1+O|î
k —> oo.
(22)
As in the case of the NLSE on a finite graph studied in [14], it can be shown that Mj satisfies the jump condition:
M,( )(xj,t,k) = M,(+)(xj,t,k)Jj(xj,t,k), k G rU ir,
(23)
where the 2 x 2 matrix Jj is defined by:
Jj
Jj(
(2)
J(4)
Jj (1),
jjJVjf,
Jj(
(3)
arg k = 0; arg k =
arg k = n; arg k = y
(24)
j
2
j
A Solution of NLSE on metric graphs and
(
J
(i)
ад
dj (k)
-Bj (k)e
e
ABj (fe) ( V dj(k)a(k)
(
J
(3) _
d(fe)
V
ABj (fe)e
-2ifcL, e2iöj
aj (k) (k)
-Bj (k) ( dj (fe)aj(fe)
a(fe) (fe)
/
J
(2)
Aßj (k) e2i0j
V «j(k)
ßj (k) e-2iöj \ (k)
1
|aj (k)|2 / (xj, t, k) — kxj + 2k t, flj (k)aj (k) + A&~®e2ifcL bj (k),
bj (k)aj (k) + A«j (fe)e2ikLj bj (k)
(k) =
ßj (k) _ _
dj (k) — Œj- (k)aj(k) - Abj (k)e2ikLj j), ¿j-(k) — a(k)aj(fe) - Aßj(k)e2ikLjbj(fe).
(25)
(26) (27)
The above expressions are the same as those for the NLSE on a finite interval, except for the bond index, j.
Following Ref. [14], one can prove Theorem. Let q0j-(x) be a smooth function. We assume that that the set of functions g0(t), gj(t), foj(t), fj(t), is admissible with respect to q0j-(x) and define the spectral functions « (k), bj(k), Aj(k), Bj(k), aj(k), bj(k) in terms of qoj(x), go(t), gij(t), foj(t), fj(t). We assume that
• « (k) has at most simple zeros, {kjn)}, for Skjra) > 0 and has no zeros for Sk = 0.
• Aj(k) has at most simple zeros, {Kjn)}, for argKjn) e (0,-^) U ^n, and has
f 1 n n 3n no zeros for arg k = 0, —, n, —.
• aj(k) has at most srmple zeros, {k<">}, for arg j e (0, n) U ^ f) and has
no zeros for arg k = 0, n, n, —. The function
dj (k) — Œj (k)Aj (k) - Abj (k)Bj (k)
(28)
has at most simple zeros, {Ajn)}, for argA(ra) e (^n) and has no zeros for n
arg k — — and arg k — n. The function
aj (k) — aj (k)aj (k) + A6(k)e2ifcLj bj (k)
(29)
1
has at most simple zeros, {vjn)}, for argvjn) G (o, ^ and has no zeros for arg k = 0,
7 ^
arg k = 2 •
None of the zeros of (k) for argk G (2n) coincides with a zero of d,(k). None of the zeros of aj(k) for arg k G ^0, ^ coincides with a zero of a(k).
None of the zeros of (k) for arg k g (°, 2 )coincides with a zero of (k) or a zero of aj (k).
7T
None of the zeros of d,(k) for arg k G (2 ,n) coincides with a zero of (k) or a
zero of aj (k).
We define Mj(xj, t,k) as the solution of the following 2 x 2 matrix RH problem: • Mj is sectionally meromorphic in c/{r U ir}, and has unit determinant.
m(-) (xj,t,k) = Mj(+)(xj,t,k)Jj(xj,t,k), k G r U ir,
(30)
where Mj is M( ) for arg k G
7T
L2
U
3n 32-,
, Mj is M7(+) for arg k G
0,
u
n,
3n 2
, and Jj is defined in terms of aj, bj, Aj, Bj, Aj, Bj, by equations (24)
and (25).
Mj-(x,t,k) = I + 0( 1 ), k ^ œ.
(31)
Let [Mj]i and [Mj]2 denote the first and the second column of the matrix Mj. Then
residue conditions:
Res [Mj(xj,t,k)]i = cj e
,(n)(1)e4i(vjn))2i+2iv(n)x
fc=v(n)
Re^[Mj (xj ,t,k)]2 = Ac: e
= Ac(™)(1)e-4i(v(n))2i-2iv(n)Xj
fc=v(n)
Res [Mj(xj,t,k)]i = cj e
(n)(2) 4i(v(n))2i+2iv(n)"
fc=A(n)
fc=A(n)
1 j (xj,
jM( x J) 2, (32)
[Mj (xj ,t,vjn))]i, (33)
[Mj (xj ,t,Ajn))]2, (34)
:[Mj (xj ,t,Ajn))]i, (35)
by
where:
c
(n)(1) _
aj(vj ))
e2»^ bj (v(n))à j (Vn))
c
(n)(2) _ ABj (Aj ))
aj (A<n))d, (A(n))
(36)
Then, Mj (xj, t,k) exists and is unique. We define qj (xj, t) in terms of Mj (xj, t,k)
qj(x, t) = 2i ■ lim k(Mj(x, t, k))12. Then, qj(x,t), together with the following functions:
v ■
x
3
e
2
j
d
qj(x0) = qoj(x), qj(0,t) = go(t) dXgj(0,t) = gj(t),
d
qj (Lj, t) = /oj(t), dxqj (Lj, t) = fij(t)
(38)
present the solution for the nonlinear Schrodinger equation (1) with initial and boundary conditions given by Eqs.(2) - (5), respectively. The proof of the theorem is similar to that of the NLSE for a finite interval treated in [14]).
Furthermore, repeating the same steps as in [14] ( for aj(k) = 1,bj(k) = 0), we get the following expressions for /1j, g1j :
= i 2k2 \m(i)(t k) go(t)
T/ij = J j) M (t, k) -
dD0
dk — / k
dD0
2 Sj (k) Aj (k)
MMji)(t,k) -
/oj(t) 2ik2
dk +
+
Aj (k)
[Fj(t, k) - Fj(t, -k)] dk, (39)
k
9D0
i
in
-gij =
2k2
dD0
Aj (k)
Mji)(t,k) -
/oj(t) 2ik2
dk k
dD0
2 Sj (k) Aj (k)
M?(i)(t,k) -
go(t) 2ik2
dk
dD0
Aj (k)
[e-2ifcLjF,-(t, k) - e2ifcLjF,-(t, -k)] dk, (40)
4
k
where:
(k) = e2ifcLj + e-2ikLj, (41)
Fj (t,k) =
«/"oj(t)
,2ifcLj j(2) _ igM MM(2) j - 2 j
+
+
L(2) - ¿A/jt)MMji) + kM(2)
2
2ifcL,'
— e
Lj2) - iA^^Mf + kMf'
Lji) - i^M(2) + kM(i) j 2 J
l(i) - i j-mj2) + kmji)
.(42)
Finally, from Eqs. (40) and (5) we obtain:
2 ^l(k)
g0(t)
dD?
Ai(k)
MM(1)dk -
dD?
2k2 Ai(k)
M(1) f01(t)
M1 - w
dk
i /M&dk - i
2i J A1(k) 2i
dD?
j=2
dD?
Sj (k) Aj (k)
dk
E
j=2
k2J)MMj(Ddk - /
Aj (k) J J Aj (k)
dD?
J 2ik2
dk
+
1 i S1(k^ 1 ^ f Sj(k) j=2 j
2U A1(k)
dD?
dk--> I dk
2i ¿W Aj (k)
"SD?
dD?
A1(k)
[e-2ifcLlF1(t, k) - e2ikLlF1(t, -k)] dk
1 i S1 (k)JL f Sj (k)
j=2 j
2U A1(k)
dD?
dk--> I . JYT"( dk
2i ¿W Aj (k)
"SD?
V i
k
Aj (k)
J=2dD?
[e-2ifcL'Fj(t, k) - e2ifcLjFj(t, -k)] dk
1 /SM dk_ / j) dk
2i/ A1(k)dk 2^7 Aj (k)dk
dD?
j=2
dD?
We note that ^j1)(0 , t,k) and ^J )(0,t,k) are solutions of:
d
—^j + 2ik2<j3;j = QQ j (0, t, k)^j,
(2)
where:
Q j (0,t,k)
V
-iA|qj (0,t)|2 d
2kqj(0, t) - i—fcj(0, t)
d
2kqj (0, t) + i—qj (0, t) iA|qj(0, t)|2
\ /
Therefore, it satisfies:
Q j (0,t,k) = a3,j SQ j (0,t,fc)E^3.
where Qj(0, t,
iA|qj (0,t)|2 d
y 2kqj(0, t) - i—qj(0, t)
d \
2kfcj(0, t) + i—fcj (0, t) ^
-iA|qj (0,t)|2
(43)
(44)
(45)
S =
(46)
A 0 \ 0A
This implies the following symmetry for boundary scattering matrix:
Sj (k) = 0*3 j SSj (fc)Sa3
(47)
k
,j
,j
where jt)=( Aj>Y 1 ( ' ^ ABj (k) Aj (k) )
5. Conclusions
In this paper, we treated the nonlinear Schrodinger equation with cubic nonlinearity on a metric graph. The boundary conditions were imposed to provide continiuty and current conservation at the graph vertex. Our approach is based an extension applied earlier by Fokas [14] for the solution to the NLSE on a finite interval with Dirichlet boundary conditions. Unlike the case of the NLSE on the interval in [14], in the case of our graph, we have:
i) Multicomponent NLSE, whose components are related to each other through the vertex boundary conditions.
ii) Additional, Neumann type boundary conditions at the graph vertex.
iii)Additional unknown functions, go, gi, fo, fi in the initial and boundary conditions. However, this doesn't lead to serious complication for adopting method of [14] for the case of graphs, although results obtained are completely different from those of NLSE for finite interval. We note that the above treatment of NLSE on star graph can be extended to other graph topologies as well.
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