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Nonlinear dynamics in the mixed state of ultraclean high-temperature superconductors
30 October 1995
_@
ELSEVIER
PHYSICS
LETTERS
A
Physics Letters A 207 (1995) 214-218
Nonlinear dynamics in the mixed state of ultraclean high-temperature superconductors Mark W. Coffey Program in Applied Mathematics, Campus Box 526, Vniversi@ of Colorado, Bouldel; CO 80309, USA’
Received 14 July 1995; revised manuscriptreceived 15 August 1995; accepted for publication Communicated
5 September
1995
by L.J. Sham
Abstract
A new formulation of the ( 1 + 1)-dimensional coupled nonlinear electrodynamics of high-temperature superconductors in the mixed state is presented. A novel nonlinear wave equation for the specific area A is derived for an ultraclean type-II superconductor. Travelling wave solutions are investigated of the nonlinear system which includes the effects of nonlocal vortex lattice elasticity and inertia. The detection of solitons could provide a method of quantitatively studying the vortex mass. An analogous dynamical system in plasma physics is described; the correspondence between characteristic length and time scales and energies is included. PACS: 41.20.Jb; 03.40.Kf; 74.60.Ge; 74.2ODe Keywords: Vortex dynamics; Korteweg-de Vries equation; Solitons
Scalar wave equation;
The dynamics of vortices in type-II superconductors [ l] is a rich field owing partly to the wide variety of forces and nonlinearities. The motion of vortices can in general influence both the electrodynamic and thermodynamic response of a superconducting system. Even at constant temperature the electrodynamic coupling of fields is nonlinear. On top of this, vortices mutually interact, through the (magnetic) Lorentz force, not at a single point, but nonlocally. Not only are the high-T, and many organic superconductors type-II, but these materials may belong to the ultraclean class with Hall force-dominated dynamics [ 21. In such materials vortex pinning and drag are negligible. Recently the dynamics of mas-
’ Current mailing address: Department of Chemistry, Regis University, 3333 Regis Boulevard, Denver, CO 80221, USA. 0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlO375-9601(95)00675-3
Travelling
wave; London theory; Nonlocal
interaction;
sive Abrikosov vortices in an ultraclean type-II superconductor was investigated in a certain weakly dispersive and nonlinear regime [ 31. The Korteweg-de Vries ( KdV) equation [ 41 was derived for the firstorder field corrections for one spatial dimension [ 31, and the Kadomtsev-Petviashvili equation [ 51 for two spatial dimensions, in the long-wavelength limit in the absence of the Hall force [ 61. Limited dissipation can be included with the use of further perturbation theory. With the inclusion of the Hall force, the dispersion relation of the linearized problem is significantly modified, and the nonlinear Schrijdinger equation appears [6]. This paper explores simple travelling wave solutions of the ( 1 + 1 )-dimensional system, and then derives a new nonlinear wave equation for the specific area A. The single fourth order equation for A, the main re-
215
M. W, Co&y/ Physics Letters A 207 (15%) 214-218
suit of this Letter, completely represents the vortex dynamics in the absence of pinning and dissipation. This equation appears to possess some remarkable properties. An analogous system has been treated by Sack and Schamel [ 71 for a cold plasma. In their system the specific volume is used and an additional (exponential) nonlin~ity occurs, which does not appear in the superconductor equations. Sack and Schamel studied ion bunching and ion wave collapse in an expanding plasma. Here a bulk, isotropic high-tem~rature superconductor is taken to be at or near absolute zero, and only frequencies well below the gap frequency are considered. In this way displacement and normal current contributions are neglected. The ~netration depth At, is taken to be independent of magnetic field, making the mesoscopic London equation linear. The detection of KdV solitons could provide a means of assessing the size, and possibly the field and temperature dependence, of the vortex mass per unit length ,U [S]. For the acoustic soliton speed varies as the square root of the ratio of the external magnetic induction to ~1.As usual for a KdV soliton, a larger amplitude pulse travels faster, with the width being proportional to the square root of the amplitude. Soliton measurements could be used to gauge the magnitude of the vortex inertial effect and possibly discriminate between contributing mechanisms. This might be done by using samples with different material parameters such as varying coherence lengths, penetration depths, critical fields, and elastic constants. The vortex mass can arise from variation of the order parameter (core contribution), generation of a local electric field, and from crystal lattice strain [ 81. It is in general derived from a complicated superposition of these sources. The mass per unit length can also be a crucial parameter in vortex tunnelling, There have been many reports of this phenomenon (see, e.g., Ref. [ 91) and perhaps the experimental techniques in this domain could be used to detect a soliton disturbance in the vortex density or the magnetic induction. In these experiments the conditions of very weak pinning and low temperature are also required. Assuming a superconductor geometry with magnetic induction B = B,, vortex velocity u = vX, and supercurrent density J = Jy, depending only on the x spatial coordinate, the governing equations are [ 31
(14 du YC=
#o SB ---t /@ax
(lb)
The magnitude of the static applied induction Bo is assumed to satisfy Bo/p~ z 2H,t, where H,i is the lower critical field. The London equation (lc) has been written in terms of the vortex area1 density n( x, t) [ IO]. On the left-hand side of the vortex equation of motion ( 1b) , ,u is the mass per unit length [ 8 1, and the ~ght-hod side is the I-orentz force #aJ, BZ. Eq. (la) expresses conservation of flux lines. By perturbing Eqs. (1) about constant vortex density Q = Ba/& and magnetic induction Bo and zero vortex velocity, with the dependence exp [i (kx - wt) 1, it is seen that the dispersion relation for the linear propagation problem is w2(k) = (~~B~/~~)~‘(l+
h2,k2)-t.
(2)
Thus for small k the dispersion relation is cubic, of the form w(k) N const x (k - iA[k3). This conclusion is an implant consequence of the nature of the nonlocal vortex interaction. It can be traced back to hi_ # 0 in Eq. (1~); for if n = B/do, the resulting dispersion relation is w2 = c%k2, where cs = &&&$$ is the acoustic wave speed. In discussing the travelling wave solutions of the system ( 1) I continue with the dimensioned form in order to make the associated energies more physically apparent. I assume a solution of the form n = n(J), u = u(l), and B = B(l), where 5 3 x - vt and V is a constant velocity. Inserting this form into Eqs. ( la) and ( 1b) and integrating with respect to 5 gives nu =ci +nY
#o &Au2-I- y-&B
(34
= /L?%+
c2*
where cl and c2 are constants of integration. The lefthand side of Eq. (3b) is made of kinetic energy per unit length of vortex and magnetic energy per unit length. Eq. (3a) expresses that in a frame moving with the wave speed, there is no change in the number of flux lines. Multiplying Eq. (lc) by the supercurrent
216
hf. W. Cofey / Physics Letters A 207 (1995~ 214-218
density, integrating with respect to I, and using Eqs. (lb) and (3a), gives
where d/dt = a/& + a?/& is the convective derivative. Eqs. (6) can be combined to yield the fourthorder equation
where es is another constant of integration. The righthand side of this equation is the difference between magnetic field energy and the kinetic energy of supercurrent, per volume. Eq. (3b), quadratic in u, can be solved for u = v( 8). The result can be inserted in Eq. (3~)) yielding a separable nonline~ first order equation for the magnetic induction. As a special case, I consider the stationary solution, V = 0. Using the boundary conditions at infinity n ---f no, B -+ Bo, and J -+ 0, then u --+ cl/no there and Eqs. (3b) and (3~) become
dtdd
d2u _ d*(nu)
c: u2 = --z+ %
(4a)
f
cxx*
d4tr
,
dt db &S?
(7)
where B has been eliminated. The main goal of this paper is to establish a novel single scalar wave equation containing the whole system (6). To do this, a new number per length variable q is introduced, q(x,t)
=
s
‘n(x’,t)dx’,
(8)
based on the continuity equation (6a). The inverse length Q and time t are taken as new independent variables and an equation for the specific area A(q, t) = 1/n( x( T, t) , t) is sought. The tr~sfo~ation back to real space is accomplished by
and
II xiri,t)
=
s
A 1$, t) dq’.
(9)
From the relations Writing B = BQ -t- 6B and expanding o(B), Eq. (4a), in the small amplitude limit, inserting the result into Eq. (4b) and integrating, yields the single soliton solution B(x)
- So =
$sech2(&$.
~~~~~~~~~~~~~~~~~~~ p(O) z lO%,lcm, the characteristic time l/w0 21 4 x lo-i3 s. These vaIues are typical of YBCO.) Then for notational ease I drop the primes to write the system (1) as
an
acnu1 =.
du
-=--9 dt
dB dx
n+d’B
ax*’
’
da dt=z
(10)
and Eqs. (6a) and (6b) follows
(5)
In accordance with standard KdV theory, the width of the soliton, cc nac,/ct, varies inversely as the square root of the amplitude. Now I perform the scaling t’ = mot, x’ = .x/h~,
z+x-
8 r3 ax =n-y*
and ~(~,t)
=
-n~(~,t).
tllb)
So far the vortex ~ntin~ty ~uation has been used, in order to define v, together with the equation of motion. Next the London ~uation,
(124 is converted to
(W
.&I-BA,
(6b)
where ” z ~3:. Finally, eli~nating duction B yields
(12b) the magnetic in-
(13)
M.W. Co@ey/ Physics Letters A 207 (I 9951214-218
Eq. ( 13) is a new formulation of the vortex dynamical system in terms of the specific area A. Small amplitude for A has not been assumed; the equation for A = A( q, t) is exact. Should a perturbation about unit area be considered, A = 1 + &A(v) exp( -iwt), it can be checked that the (scaled form of) dispersion relation (2) follows, under the assumption d2SA/dq2 = -@??A. Eq. (13) could be used for following vortex displacements until hysteretic effects appear. This happens roughly when a small reciprocal-integer multiple of the intervortex spacing is reached. Eq. ( 13) might be viewed physically as a nonlinear ~uation of motion for A, with an associated time-dependent potential. In this context, there is an acceleration-dependent force. For smaller values of A, the vortices are closer together and so interact more strongly. The repulsive (Lorentz) force tends to spread vortices apart, decreasing A and thereby, given fluxline conservation, allowing the possibility of an oscillatory motion. A preliminary Painlevd analysis [ 111 has been performed of the nonlinear wave equation for A, written in the form A4A,, + ( 1 - A,,) (3A; - AA,,) f 3AA,Att9
- A2A,rv,, = 0.
(14)
With A(q r) - Ao( v, t) c#P(9, t) at leading order, the singul~ity exponent is LY= -1, with most singular term with #-7 and Ao( q, t) = 12& (9, t). Resonances are found to occur at r = 2 (repeated) and r = 4. The Painleve test is not passed, for (i) the repeated resonance generally introduces a logarithmic term, and (ii) there remains a constraint on the singularity manifold 4. However, this compatibility condition,
4:4?pl + 4;4rt - 24,14t4r?? = 0,
(15)
is of a special trilinear form. Besides an exponential solution, Eq. (15) is satisfied by any function (6(q -4. Vf) of travelling wave form. In fact Eq. f 15) is the Bateman equation [ 12,131, which is completely integrable. Its general solution is given implicitly as ?Ifl(4)
+f.f2(+)
=cs
(16)
where fi and f:! are arbitrary twice differentiable functions and the constant c may generally be taken as
217
zero or one. Although a truncated Painlevt expansion for A cannot be expected to yield a general solution, it could still be useful in obtaining special solutions, Eq. (13) has been derived in the continuum approximation for the vortex density n [ lo]. It accounts for vortex inertia and nonlocal interaction but neglects drag, pinning, and the Hall force. This regime may be close to that considered in vortex tunnelling [ 2,9]. An analogous dynamical system exists in plasma physics in ion acoustic waves in the small-amplitude limit [ 41. The correspondences for the dependent variables are n +-+ni, the volume density of ions, u ++ vi, the ion velocity, and B ++ 4 + 1, where 4 is the electrostatic potential. The magnetic (Lorentzf force in the vortex lattice is replaced with the electric force in the two-component plasma. Therefore the current density in the superconductor plays a role analogous to the electric field in the plasma. Concerning the constant coefficients in the respective equations of motion, p cf mi, the ionic mass, and ~o/~ H e, the electronic charge. The characteristic length and time scales correspond as XL H L, the electronic Debye shielding length, and wa ts Opi, the ion plasma frequency. As seen from Eqs. (3), the vortex kinetic energy per unit length corresponds to the ionic kinetic energy, the supercurrent kinetic energy density corresponds to the electric field energy density, and the magnetic field energy per unit length ( ~o/~o) B corresponds to the electrostatic potential energy e#. The soliton speed c, is parallel to the AlfvCn speed U.&. When electron inertia is neglected in the plasma, an integration can be performed to give the electron number density explicitly in terms of the electrostatic potential. This additional nonlinearity does not occur in the vortex dynamics equations. I ignored magnetic field nonline~ity in the pene~ation depth so that the London equation ( lc) is linear. This approximation is well justified for a very wide range of field for highT, superconductors owing to their very large upper critical fields. The weak nonlinearities in the present model include bilinearity in the vortex continuity Eq. ( la) and convective differentiation in the equation of motion (lb). The use of the continuum density n allows the modelling of nonlocal vortex elasticity, over a characteristic distance of AL, including tilt and compression modes of the lattice. The nonlocal vortex interaction is crucial in obtaining the long-wave cubic dispersion
218
M.W. @fey/
Physics letters A 207 (1995) 214-218
relation, from Eq. (2). It is possible to consider two-dimensional (in space) generali~tions of the equations presented here, the four dependent variables being n, u,, ur, and B = B,. The dispersion relation is still of the form of E$ (2), with k2 z k’, + k$ The KadomtsevPetviashvili equation, a 2D generalization of the KdV equation, has been obtained [6]. Analysis of the 2D bulk problem, with and without Hall force term, is presented elsewhere [ 61. I thank S. Chakravarty and T. Werckman for useful discussions. This work was partially supported by the Air Force Office of Scientific Research under grant No. F49620-940 120. References ill
D. Saint-James, G. Sarma and E.J. Thomas, Type II su~rconductiv~ty (Pergamon, Oxford, 1969) ; R.P. Huebener, Magnetic flux stmctures in superconductors (Springer, Berlin, 1979) : R.D. Parks, ed., Superconductivity (Dekker, New York, 1969).
[2] G. Blat&r, V.B. Geshkenbein and V.M. Vinokur, Phys. Rev. Lett. 66 (1991) 3297. (31 M.W. Coffey, Phys. Rev. B 52 ( 1995), to appear. [4] G.L. Lamb Jr., Elements of soliton theory (Wiley, New York, 1980); R.K. Dodd et al., Solitons and nonlinear wave equations (Academic Press, New York, 1982). [5] B.B. Kadomtsev and V.I. Petviashvili, Sov. Phys. Dokl. 15 (1970) 539. [6] M.W. Coffey, unpublished (1995). [ 71 Ch. Sack and H. Schamel, Phys. L.&t. A 110 f 1985) 206; Plasma Phys. Contr. Fusion 27 (1985) 717. [S] H. Suhl, Phys. Rev. Lett. 14 ( 1965) 226; M.W. Coffey, J. Low Temp. Phys. 96 (1994) 81; 98 f 1995) 159; M.W. Coffey and Z. Hao, Phys. Rev. B 42 (1991) 5230. [9] N. Giordano, Phys. Rev. L&t. 61 (1988) 2137; AC. Mota et al., Physica C 185-189 (1991) 343; Y. Liu, D.B. Haviland, L. Glazman and A.M. Goldman, Phys. Rev. Len. 68 ( 1992) 2224. (IO] M.W. Coffey and J.R. Clem, Phys. Rev. Len. 67 (1991) 386; Phys. Rev. B 45 (1992) 9872; 46 (1992) 11757. [ 111 W. Heremau and E. van den Buick, in: Finite dimensional integrable nonlinear dyn~ical systems, eds. P.G.L. Leach and W.H. Steeb, World Scientific (1988) p. 117. [ 121 H. Bateman, Proc. R. Sot. A 125 (1929) 598. [ 131 P.R. Gambedian, Partial differential equations (Wiley, New York, 1964).
_@
ELSEVIER
PHYSICS
LETTERS
A
Physics Letters A 207 (1995) 214-218
Nonlinear dynamics in the mixed state of ultraclean high-temperature superconductors Mark W. Coffey Program in Applied Mathematics, Campus Box 526, Vniversi@ of Colorado, Bouldel; CO 80309, USA’
Received 14 July 1995; revised manuscriptreceived 15 August 1995; accepted for publication Communicated
5 September
1995
by L.J. Sham
Abstract
A new formulation of the ( 1 + 1)-dimensional coupled nonlinear electrodynamics of high-temperature superconductors in the mixed state is presented. A novel nonlinear wave equation for the specific area A is derived for an ultraclean type-II superconductor. Travelling wave solutions are investigated of the nonlinear system which includes the effects of nonlocal vortex lattice elasticity and inertia. The detection of solitons could provide a method of quantitatively studying the vortex mass. An analogous dynamical system in plasma physics is described; the correspondence between characteristic length and time scales and energies is included. PACS: 41.20.Jb; 03.40.Kf; 74.60.Ge; 74.2ODe Keywords: Vortex dynamics; Korteweg-de Vries equation; Solitons
Scalar wave equation;
The dynamics of vortices in type-II superconductors [ l] is a rich field owing partly to the wide variety of forces and nonlinearities. The motion of vortices can in general influence both the electrodynamic and thermodynamic response of a superconducting system. Even at constant temperature the electrodynamic coupling of fields is nonlinear. On top of this, vortices mutually interact, through the (magnetic) Lorentz force, not at a single point, but nonlocally. Not only are the high-T, and many organic superconductors type-II, but these materials may belong to the ultraclean class with Hall force-dominated dynamics [ 21. In such materials vortex pinning and drag are negligible. Recently the dynamics of mas-
’ Current mailing address: Department of Chemistry, Regis University, 3333 Regis Boulevard, Denver, CO 80221, USA. 0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlO375-9601(95)00675-3
Travelling
wave; London theory; Nonlocal
interaction;
sive Abrikosov vortices in an ultraclean type-II superconductor was investigated in a certain weakly dispersive and nonlinear regime [ 31. The Korteweg-de Vries ( KdV) equation [ 41 was derived for the firstorder field corrections for one spatial dimension [ 31, and the Kadomtsev-Petviashvili equation [ 51 for two spatial dimensions, in the long-wavelength limit in the absence of the Hall force [ 61. Limited dissipation can be included with the use of further perturbation theory. With the inclusion of the Hall force, the dispersion relation of the linearized problem is significantly modified, and the nonlinear Schrijdinger equation appears [6]. This paper explores simple travelling wave solutions of the ( 1 + 1 )-dimensional system, and then derives a new nonlinear wave equation for the specific area A. The single fourth order equation for A, the main re-
215
M. W, Co&y/ Physics Letters A 207 (15%) 214-218
suit of this Letter, completely represents the vortex dynamics in the absence of pinning and dissipation. This equation appears to possess some remarkable properties. An analogous system has been treated by Sack and Schamel [ 71 for a cold plasma. In their system the specific volume is used and an additional (exponential) nonlin~ity occurs, which does not appear in the superconductor equations. Sack and Schamel studied ion bunching and ion wave collapse in an expanding plasma. Here a bulk, isotropic high-tem~rature superconductor is taken to be at or near absolute zero, and only frequencies well below the gap frequency are considered. In this way displacement and normal current contributions are neglected. The ~netration depth At, is taken to be independent of magnetic field, making the mesoscopic London equation linear. The detection of KdV solitons could provide a means of assessing the size, and possibly the field and temperature dependence, of the vortex mass per unit length ,U [S]. For the acoustic soliton speed varies as the square root of the ratio of the external magnetic induction to ~1.As usual for a KdV soliton, a larger amplitude pulse travels faster, with the width being proportional to the square root of the amplitude. Soliton measurements could be used to gauge the magnitude of the vortex inertial effect and possibly discriminate between contributing mechanisms. This might be done by using samples with different material parameters such as varying coherence lengths, penetration depths, critical fields, and elastic constants. The vortex mass can arise from variation of the order parameter (core contribution), generation of a local electric field, and from crystal lattice strain [ 81. It is in general derived from a complicated superposition of these sources. The mass per unit length can also be a crucial parameter in vortex tunnelling, There have been many reports of this phenomenon (see, e.g., Ref. [ 91) and perhaps the experimental techniques in this domain could be used to detect a soliton disturbance in the vortex density or the magnetic induction. In these experiments the conditions of very weak pinning and low temperature are also required. Assuming a superconductor geometry with magnetic induction B = B,, vortex velocity u = vX, and supercurrent density J = Jy, depending only on the x spatial coordinate, the governing equations are [ 31
(14 du YC=
#o SB ---t /@ax
(lb)
The magnitude of the static applied induction Bo is assumed to satisfy Bo/p~ z 2H,t, where H,i is the lower critical field. The London equation (lc) has been written in terms of the vortex area1 density n( x, t) [ IO]. On the left-hand side of the vortex equation of motion ( 1b) , ,u is the mass per unit length [ 8 1, and the ~ght-hod side is the I-orentz force #aJ, BZ. Eq. (la) expresses conservation of flux lines. By perturbing Eqs. (1) about constant vortex density Q = Ba/& and magnetic induction Bo and zero vortex velocity, with the dependence exp [i (kx - wt) 1, it is seen that the dispersion relation for the linear propagation problem is w2(k) = (~~B~/~~)~‘(l+
h2,k2)-t.
(2)
Thus for small k the dispersion relation is cubic, of the form w(k) N const x (k - iA[k3). This conclusion is an implant consequence of the nature of the nonlocal vortex interaction. It can be traced back to hi_ # 0 in Eq. (1~); for if n = B/do, the resulting dispersion relation is w2 = c%k2, where cs = &&&$$ is the acoustic wave speed. In discussing the travelling wave solutions of the system ( 1) I continue with the dimensioned form in order to make the associated energies more physically apparent. I assume a solution of the form n = n(J), u = u(l), and B = B(l), where 5 3 x - vt and V is a constant velocity. Inserting this form into Eqs. ( la) and ( 1b) and integrating with respect to 5 gives nu =ci +nY
#o &Au2-I- y-&B
(34
= /L?%+
c2*
where cl and c2 are constants of integration. The lefthand side of Eq. (3b) is made of kinetic energy per unit length of vortex and magnetic energy per unit length. Eq. (3a) expresses that in a frame moving with the wave speed, there is no change in the number of flux lines. Multiplying Eq. (lc) by the supercurrent
216
hf. W. Cofey / Physics Letters A 207 (1995~ 214-218
density, integrating with respect to I, and using Eqs. (lb) and (3a), gives
where d/dt = a/& + a?/& is the convective derivative. Eqs. (6) can be combined to yield the fourthorder equation
where es is another constant of integration. The righthand side of this equation is the difference between magnetic field energy and the kinetic energy of supercurrent, per volume. Eq. (3b), quadratic in u, can be solved for u = v( 8). The result can be inserted in Eq. (3~)) yielding a separable nonline~ first order equation for the magnetic induction. As a special case, I consider the stationary solution, V = 0. Using the boundary conditions at infinity n ---f no, B -+ Bo, and J -+ 0, then u --+ cl/no there and Eqs. (3b) and (3~) become
dtdd
d2u _ d*(nu)
c: u2 = --z+ %
(4a)
f
cxx*
d4tr
,
dt db &S?
(7)
where B has been eliminated. The main goal of this paper is to establish a novel single scalar wave equation containing the whole system (6). To do this, a new number per length variable q is introduced, q(x,t)
=
s
‘n(x’,t)dx’,
(8)
based on the continuity equation (6a). The inverse length Q and time t are taken as new independent variables and an equation for the specific area A(q, t) = 1/n( x( T, t) , t) is sought. The tr~sfo~ation back to real space is accomplished by
and
II xiri,t)
=
s
A 1$, t) dq’.
(9)
From the relations Writing B = BQ -t- 6B and expanding o(B), Eq. (4a), in the small amplitude limit, inserting the result into Eq. (4b) and integrating, yields the single soliton solution B(x)
- So =
$sech2(&$.
~~~~~~~~~~~~~~~~~~~ p(O) z lO%,lcm, the characteristic time l/w0 21 4 x lo-i3 s. These vaIues are typical of YBCO.) Then for notational ease I drop the primes to write the system (1) as
an
acnu1 =.
du
-=--9 dt
dB dx
n+d’B
ax*’
’
da dt=z
(10)
and Eqs. (6a) and (6b) follows
(5)
In accordance with standard KdV theory, the width of the soliton, cc nac,/ct, varies inversely as the square root of the amplitude. Now I perform the scaling t’ = mot, x’ = .x/h~,
z+x-
8 r3 ax =n-y*
and ~(~,t)
=
-n~(~,t).
tllb)
So far the vortex ~ntin~ty ~uation has been used, in order to define v, together with the equation of motion. Next the London ~uation,
(124 is converted to
(W
.&I-BA,
(6b)
where ” z ~3:. Finally, eli~nating duction B yields
(12b) the magnetic in-
(13)
M.W. Co@ey/ Physics Letters A 207 (I 9951214-218
Eq. ( 13) is a new formulation of the vortex dynamical system in terms of the specific area A. Small amplitude for A has not been assumed; the equation for A = A( q, t) is exact. Should a perturbation about unit area be considered, A = 1 + &A(v) exp( -iwt), it can be checked that the (scaled form of) dispersion relation (2) follows, under the assumption d2SA/dq2 = -@??A. Eq. (13) could be used for following vortex displacements until hysteretic effects appear. This happens roughly when a small reciprocal-integer multiple of the intervortex spacing is reached. Eq. ( 13) might be viewed physically as a nonlinear ~uation of motion for A, with an associated time-dependent potential. In this context, there is an acceleration-dependent force. For smaller values of A, the vortices are closer together and so interact more strongly. The repulsive (Lorentz) force tends to spread vortices apart, decreasing A and thereby, given fluxline conservation, allowing the possibility of an oscillatory motion. A preliminary Painlevd analysis [ 111 has been performed of the nonlinear wave equation for A, written in the form A4A,, + ( 1 - A,,) (3A; - AA,,) f 3AA,Att9
- A2A,rv,, = 0.
(14)
With A(q r) - Ao( v, t) c#P(9, t) at leading order, the singul~ity exponent is LY= -1, with most singular term with #-7 and Ao( q, t) = 12& (9, t). Resonances are found to occur at r = 2 (repeated) and r = 4. The Painleve test is not passed, for (i) the repeated resonance generally introduces a logarithmic term, and (ii) there remains a constraint on the singularity manifold 4. However, this compatibility condition,
4:4?pl + 4;4rt - 24,14t4r?? = 0,
(15)
is of a special trilinear form. Besides an exponential solution, Eq. (15) is satisfied by any function (6(q -4. Vf) of travelling wave form. In fact Eq. f 15) is the Bateman equation [ 12,131, which is completely integrable. Its general solution is given implicitly as ?Ifl(4)
+f.f2(+)
=cs
(16)
where fi and f:! are arbitrary twice differentiable functions and the constant c may generally be taken as
217
zero or one. Although a truncated Painlevt expansion for A cannot be expected to yield a general solution, it could still be useful in obtaining special solutions, Eq. (13) has been derived in the continuum approximation for the vortex density n [ lo]. It accounts for vortex inertia and nonlocal interaction but neglects drag, pinning, and the Hall force. This regime may be close to that considered in vortex tunnelling [ 2,9]. An analogous dynamical system exists in plasma physics in ion acoustic waves in the small-amplitude limit [ 41. The correspondences for the dependent variables are n +-+ni, the volume density of ions, u ++ vi, the ion velocity, and B ++ 4 + 1, where 4 is the electrostatic potential. The magnetic (Lorentzf force in the vortex lattice is replaced with the electric force in the two-component plasma. Therefore the current density in the superconductor plays a role analogous to the electric field in the plasma. Concerning the constant coefficients in the respective equations of motion, p cf mi, the ionic mass, and ~o/~ H e, the electronic charge. The characteristic length and time scales correspond as XL H L, the electronic Debye shielding length, and wa ts Opi, the ion plasma frequency. As seen from Eqs. (3), the vortex kinetic energy per unit length corresponds to the ionic kinetic energy, the supercurrent kinetic energy density corresponds to the electric field energy density, and the magnetic field energy per unit length ( ~o/~o) B corresponds to the electrostatic potential energy e#. The soliton speed c, is parallel to the AlfvCn speed U.&. When electron inertia is neglected in the plasma, an integration can be performed to give the electron number density explicitly in terms of the electrostatic potential. This additional nonlinearity does not occur in the vortex dynamics equations. I ignored magnetic field nonline~ity in the pene~ation depth so that the London equation ( lc) is linear. This approximation is well justified for a very wide range of field for highT, superconductors owing to their very large upper critical fields. The weak nonlinearities in the present model include bilinearity in the vortex continuity Eq. ( la) and convective differentiation in the equation of motion (lb). The use of the continuum density n allows the modelling of nonlocal vortex elasticity, over a characteristic distance of AL, including tilt and compression modes of the lattice. The nonlocal vortex interaction is crucial in obtaining the long-wave cubic dispersion
218
M.W. @fey/
Physics letters A 207 (1995) 214-218
relation, from Eq. (2). It is possible to consider two-dimensional (in space) generali~tions of the equations presented here, the four dependent variables being n, u,, ur, and B = B,. The dispersion relation is still of the form of E$ (2), with k2 z k’, + k$ The KadomtsevPetviashvili equation, a 2D generalization of the KdV equation, has been obtained [6]. Analysis of the 2D bulk problem, with and without Hall force term, is presented elsewhere [ 61. I thank S. Chakravarty and T. Werckman for useful discussions. This work was partially supported by the Air Force Office of Scientific Research under grant No. F49620-940 120. References ill
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