kink energy sum rule in a two-component scalar field model of 1 + 1 dimensions

Volume 112A, number 3,4 PHYSICS LETTERS 21 October 1985

KINK ENERGY SUM RULE IN A T W O - C O M P O N E N T SCALAR FIELD MO D EL OF 1 + 1 DIMENSIONS

Hiroyuki ITO

Department of Physics, Faculty of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan

Received 2 July 1985; accepted for publication 15 August 1985

A special kink energy sum rule is proven rigorously in a two-component scalar field model of 1 + 1 dimensions. It has been previously examined only numerically. The proof is obtained by using the correspondence to an integrable hamiltonian system with two degrees of freedom.

Novel properties of solitons in nonlinear systems have been often explained by the concept of complete integrability. We will show that a special kink energy sum rule exists in a coupled relativistic scalar field model of 1 + 1 dimensions due to more general properties of the system.

The model, first introduced by Montonen [1 ] and Sarker, TruUinger and Bishop [2], has the lagrangian

J d x I • 1 = [~(q~ + 4 2 ) _ ~(q~2 + q~2)_ V ( q l , q 2 ) ] , L

(1)

where (ql, q2) are two real scalar fields, the prime and dot denote a x and Ot, respectively. The local potential is given by

1 2 2 1 2 V(ql , q 2 ) = _ ] ( q 1 +q2)+~(q 1 +q2)2 • i 2 2 • ~o q2 ' (2)

where the first two terms represent a rotated ~b 4 potential and the last term (with the anisotropy parameter o: real positive constant) breaks the continuous rotational symmetry of the potential. In this study, we restrict ourselves to time-independent kink solutions. Due to the relativistic invariance of the lagrangian, moving solutions with the velocity 0 (Iv[ < 1) are trivially obtained by transforming to a moving coordinate frame s = 7(x - ot), with 3' - (1 - 02) -1/2.

The Euler-Lagrange equation for the coupled

0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

fields is given by

d2qi/dx 2 = 0 V(q 1, q2)/aqi (i = 1 ,2 ) . (3)

Since the system has doubly degenerate ground states at (ql, q2) = (-+1,0), we expect two types of kink ex- citations. One is the topological kink, which connects the two ground states and the other is the non-topological kink, which connects one of the ground states with itself. If a kink solution (qK(x) , qK(x ) ) is obtained, the excitation energy of this kink is given by

EK(O = O) = f dx [g(qlK(x), qK(x ) ) -- g 0 ] , (4)

where the hamiltonian density is given by

g = i[.qll~ ,2 +q~2) + V ( q l , q 2 ) , (5)

and g o = V(q l = + 1 ,q2 = O) - V 0 ( - -1 /4) is the local energy at the ground state. A travelling kink solution with the velocity 0 has the excitation energy eK(V ) = -IEK(0 ).

Three types of kink solutions have been obtained in analytic or numerical forms [3,4] (see fig. 1). The one-component topological kink solution (TK1) with the excitation energy E TK 1(0) = 2x/~-/3 was first introduced [5]. This solution exists in the whole region o > 0, and the trajectory in the (ql, q2) plane is the straight line connecting the two ground states. The two-component topological kink solution (TK2) and its mirror image (TK2*) [1,2] exist only for 0 < o < 1, with the excitation energy E TK 2(0) = E TK 2* (0)

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q2

~ q Fig. 1. Trajectories of topological kink solutions and some non-topological kink solutions of the model with o = 1/2.

H=½(q{2 ,2 + q2 ) + U ( q l , q 2 ) " (7)

Very recently the second algebraic integral of this hamiltonian system has been found by Hietarinta [11 ]. The requirement of the finiteness of the kink excitation energy demands

Iql(+Oo)l = 1 , q 2 ( - +°°) = 0 ,

q ~(+oo) = q~(+oo) = 0 . (8)

The action of this dynamical system is defined by

X

S( { qi(x) }, x) = f dx L(q 1 (x), q2(x))

= x/2-o(1 - 02/3). The trajectories in the (q l, q 2) plane are the upper half (q2/> 0) of the ellipse connecting the two ground states (TK2) and the lower half (TK2*). respectively. The analytic expression of the two-component non-topological kink solution (NTK) has been obtained at o = 1/2 by Rajaraman [6]. The trajectory in the (ql, q2) plane is a circle connecting one of the ground states with itself, symmetric with the q l. axis, and the excitation energy is ENTK(0) = 9X/2/8. The curious energy sum rule has been pointed out at o = 1/2 [7],

ENTK(0 ) = ETK 1(0) + E TK 2(0) . (6)

Subbaswamy and Trullinger [3] have shown by numerical calculations that the non-topological kink solutions with the same topology exist even for other values of a in the region 0 < cr < 1, and that the energies of kinks also satisfy the sum rule (6). Further- more, Magyari and Thomas [4] have recently shown, by numerical integration, that there exist infinitely many non-topological kink solutions, which pass the special point (ql , q2) = (o, 0) [or ( - o , 0) for the solutions related to the other ground state[. Surprising- ly, by their numerical evaluations, these solutions have been found to have the same energy, so that the sum rule (6) has general validity in this model. We will give the proof of this special sum rule in this paper.

If we think of the variable x as " t ime" and the two fields (q 1 (x), q2(x)) as the coordinates of a unit-mass point particle in two-dimensional space, eq. (3) can be regarded as Newton's equation for this particle in a potential U(q 1, q2) ~ -V(ql , q2) [8 -10] . The hamiltonian of this dynamical system is

X

= f dx [~(q~2 + q~2) _ u(ql ' q2)] , (9)

where L is the lagrangian. Comparing this expression with the definition of the excitation energy of the kink solution, (4), we find a relation

fax Vo- f f Vo,(lO)

where S is the total action. The calculation of the kink energies in the field theory just corresponds to the calculation of the kink-trajectory dependence of the total action in the dynamical system. To deter- mine the action, we use the Hamil ton-Jacobi method [12]. We introduce a new coordinate system (elliptical) polar coordinates) (~t, v) [3,13], by

ql = ° c ° s h u c ° s v , q2 = ° s i n h u s i n v , (11)

with 0 ~< u < oo and - ~ < v ~ ~. The hamiltonian (7) becomes

H(u, v, Pu' Pv)

2 2 a(u) + b(v) Pu + Pu +

2o2(cosh2u - cos2v) cosh2u - cos2v

where the momenta are

Pu = 02( c°sh2u - c°s2v) u ' ,

Po = a2( c°sh2u - c°s2°) ° ' ,

(12)

(13)

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and a(u) and b(o) are given by

a(u) = -~ 02 { 2 cosh2u - (2 +02) cosh4u + o2cosh6/a } ,

b(u) = ¼o2(2 cos2u - (2 + 02) cos4o + o2cos6u} • (14)

The Hamilton-Jacobi equation,

aS/Ox + H(u, v, aS/bu, 3S/ao) = O, (15)

is separable in variables and can be solved as

S(u, o, x ) = S 1 (u ) + S2 (o ) - EX , (16)

with

U

= sgn(u') f du sinh Sl(U) U

X {2o2E + I F - 2o2a(u)]/(cosh2u- 1)} 1/2 ,

l)

S2(v ) = sgn(v') f dv [sin ol

× {2o2E - IF+ 2o2b(v)]/(1 - c o s 2 o ) } 1/2 , (17)

where E and F are constants of separation. The condi- tion (8) determines them to be E = 1/4 and F = 0. Then eq. (17) is simplified as

3 u Sl(U ) = sgn(u')~22 f du sinh u I1/o 2 - cosh2ul ,

, 0 3 o S2(o ) = sgn(v )~--~ f do Isino[ [1/o 2 - cos2o] .(18)

The equation for trajectories is obtained by

aS~OF = const. (19)

Then an equation for the trajectory flow in the (u, v) plane is obtained as

du/dv = sgn(u') sgn(o') sgn(v)

X (sinh u[ 1/o 2 - coshEu [)/(sin v[ 1/o 2 - cos2v[). (20)

For 0 < o < 1, the flow du/do becomes indefinite at some points.

(i) ground states

A+ : (u, u) = (Umax, 0 +) , B± : (Umax, z'Tr), (21)

which correspond to the two ground states (ql, q2) = (1,0) and ( -1 ,0 ) , respectively, and Urea x -= cosh-l(1/o).

(ii) foci

Fl~: (u, o) = (0, 0+), F~: (0, +Tr), (22)

which correspond to the two foci of the elliptical- polar coordinate system, i.e., (ql, q2) = (0, 0) and ( -0 , 0), respectively. We easily find that the following kink trajectories, which connect the above points, are the solutions of eq. (20). Straight lines B+-A+ and B _ - A _ in the (u, u) plane correspond to the kink trajectories TK2 and TK2*, respectively. And lines B . - F ~ - F ~ - A + and B _ - F ~ - - F I ~ - A _ correspond to the other kink trajectory TK1. In fig. 2, these kink trajectories are depicted in the (u, o) plane. The anal- ysis of the trajectory flow shows that the kink trajectories TK2 and TK2* are not crossed by other trajectories, therefore these trajectories may be called the "separatrices" which separate the closed trajectories and divergent ones. Eq. (1 9) given the trajectories of the NTK solutions

[cosh u + sgn(u')] [1/o - Sgn(u') cosh u] ° d

[cosh u - sgn(u')] [1]o + sgn(u') coshu] °

_ [1 + sgn(v') sgn(v) cos v] [1 - sgn(u') sgn(v) cos v]

[ l / o - sgn(u') sgn(v) cos v] a × [11o + sgn(v') sgn(v) cos v] o ' (23)

where d (>~O)is the parameter which is determined by the constant in eq. (19) and characterizes the NTK solutions. In fig. 2 the typical NTK trajectories (B+

B_) are depicted for three values o f d with o = 1/2. The anti-kink trajectories (B_ ~ B+) and the trajectories or solutions related to the other ground state (A±) are obtained from the trajectories (B+ ~ B_) by some symmetry operations, which can be easily derived from eq. (23). We summarize the general nature of the NTK trajectories.

(1) For any value of d, trajectory passes the focus F R (or FL). It approaches FI~ , keeping the gradient du/ dv= alx/-d.

(2) Two trajectories with the parameters d = • and d = 1/r form twin-trajectories, which are related to

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+ - - U M A X

-m 0 71: + +

F L FR F R F L

Fig. 2. Kink trajectories for TK1 (I+, I_), TK2 (II+) and TK2* (II_) are shown. NTK trajectories for three values o f d with o = 1/2 are also depicted [solid line (d = 10), broken line (d = 0.1), and dash-dotted line (d = 1). These trajectories are depicted also in the

(ql,q2) plane in fig. 1.

each other by a reflection at the u-axis. For the special case with d = 1, the twin-trajectories become a single one, which is symmetric with the u-axis. This special trajectory corresponds to the solution found by Rajaraman [6], and Subbaswamy and Trullinger [3].

As the anisotropy parameter o increases to 1, the value of Uma x decreases to 0, then the allowed region for trajectories becomes smaller, and at o = 1 the trajectories are confined on the one-dimensional region (on v-axis). For o >/1, the only allowed trajectory is the one-dimensional motion (TK1) in the region cos- l (1/o) <~ Ivl ~< 7r- cos- l (1/o) and u -= 0.

Now, we calculate the kink excitation energies for various solutions with the help of the relation in eq. (10). Using eqs. (10) and (16) and noting E = - V 0 (= 1/4), we obtain a more direct relation,

EK(0 ) =S 1 +S 2 , (24)

where S 1 and S 2 are the separated total actions, which are given by eq. (18), the integrals being performed along the kink trajectory in the (u, o) plane. For the TK2 (TK2*) solution, the integration along the cor- responding trajectory B+-A+ ( B _ - A _ ) gives ETK2(0) = ETK2*(0) = X/~ o(1 -- o2/3) = C 2 . The

+ + (FI~ - F R) also gives integration along FL--F R the same contribution for the total action. Moreover

if we define the contribution from the line B+--F~_ (also from lines FI~-A+, B_-FI~ , and FI~-A_ ) as C 1 [= (2/3 - o + 0 3 / 3 ) ~ ] , then the kink energy for the TK1 solution is calculated as E TK 1(0) = 2C 1 + C 2 -'- 2x/2-/3. Because C 1 > 0 in the region 0 < o < 1, the energy ETK 1 is always larger than the energyE TK2. And finally for the NTK solutions, since the action can be separated in variables and all NTK trajectories pass the focus F R (or FL), the kink energies are independent of the shapes of the trajectories and can be obtained by the simple summation

ENTK(0) = 2C 1 + 2C 2 = V/2(2/3 + 03/3). (25)

We have thus proved the kink energy sum rule (6). The existence of this special sum rule is due to the

separability of the Hamilton-Jacobi equation, which immediately leads to the integrability of this hamiltonian system, and the special topological property of kink trajectories. The TK1 and all NTK trajectories pass the focus F R (or FL) and the TK2 (TK2*) does not. This fact leads to the conclusion that the TK1 and all NTK solutions are linearly unstable, and on the other hand, the TK2 (TK2*) solution is stable. This conclusion can be obtained from the application of the Morse index theorem in the calculus of varia- tions in the large to the stability problem of the kink

122


solutions in the 1 + 1-dimensional coupled nonlinear Kle in-Gordon type equations [14].

I am grateful to Professor Y. Wada for critical read- ings of the manuscript and useful discussion. I also express my sincere thanks to Dr. Y. Ono, Dr. S. Takesue, H. Tasaki and Y. Ohfuti for stimulating discussion.

Reference$

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(1981) 379. [ 4] E. Magyari and H. Thomas, Phys. Lett. 100A (1984) 11.

[5] R. Rajaraman and E.J. Weinberg, Phys. Rev. Dll (1975) 2950.

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(1980) 1495. [8] N.H. Christ and T.D. Lee, Phys. Rev. D12 (1975)

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kink energy sum rule in a two-component scalar field model of 1 + 1 dimensions

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