Dresner, Stability of superconductors.2002
.pdf
200 |
Appendix A |
c´ =λac
t´ = λbt
z´ = λ z
0 < λ <
8
(A.1.2)
Here l is the group parameter that labels individual transformations of a group and a and b are parameters that label groups of the family. The word “invariance” means that if we imagine the partial differential equations in Eq. (A.1.1) written in terms of the primed variables and then substitute the equations in Eqs. (A.1.2), the powers of l thereby introduced cancel, and we recover the original partial differential equation again, this time written in terms of the unprimed variables. In general, for the powers of l to cancel, the parameters a and b must obey a linear equation of constraint
Ma + Nb = L |
(A.1.3) |
where the coefficients M, N, and L depend on the form of the partial differential equation. For the partial differential equations (1), (2), and (3) above, we find (M,N,L ) = (n–1,1,2), (1,–1,–2), and (2,–3,–4), respectively.
A.2. SIMILARITY SOLUTIONS
If c(z,t) is a solution of the invariant partial differential equation, so is c´(z´,t´) since the partial differential equation in the primed variables is the same as the partial differential equation in the unprimed variables. In general, the transformed solution c´(z´,t´) is not the same as its pre-image c(z,t). But there may be some solutions that are carried into themselves under transformation by one group of the family (i.e., invariant to the transformations (A.1.2) for all l but for a particular choice of a and b). Solutions of the form
c = t a/by(z/t1/b) |
(A.2.1) |
where y is a function of the single variable x = z/t1/ b, have this property. They are the similarity solutions. It can be shown (Cohen, 1931) that all invariant solutions must have the form (A.2.1).
When we substitute Eq. (A.2.1) into the partial differential equation, we obtain an ordinary differential equation for the function y(x) since it is a function of only one variable. I call this ordinary differential equation the principal ordinary differential equation.
The reader probably realizes that the form (A.2.1) restricts the boundary and initial conditions the similarity solutions fulfill. For example, c(0,t) = y(0)t a/b, a power law in t. Furthermore, if c(∞,t)= 0, then y(∞) = 0, so that then c(z,0) = 0 also.
The Method of Similarity Solutions |
201 |
Different choices of a and b (that fulfill the constraint A.1.3) may lead to different boundary and initial conditions and thus define different problems, but the form A.2.1 is still rather limiting.
A.3. THE ASSOCIATED GROUP
The form of the principal ordinary differential equation depends, naturally, on the form of the partial differential equation, but it can be proved (Dresner, 1983) that the principal ordinary differential equation, whatever its form, is invariant to the stretching group
y´ =mL/My |
|
0 < m < 8 |
(A.3.1) |
x´ = mx
I call this group the associated group.
The usefulness of the associated group rests on a theorem of Sophus Lie’s (Cohen, 1931; Dresner, 1983), according to which we can reduce the order of the principal ordinary differential equation by one if we use as a new independent variable an invariant of the group and as a new dependent variable a first differential
invariant. An invariant is any function f (x,y) that is unchanged under the transformations A.3.1, i.e, any function f(x,y) for which f(x,y) = f(x´,y´) = f (mx,mL/M y). The function u = yx–L/M is such an invariant, and it can be shown that the most general
invariant is a function of u. A first differential invariant is any function g(x,y,ý) (here ý = dy/dx) such that g(x,y,ý) = g(x´,y´,ý´) = g(mx,mL/My , L/M–1ý). (N.B.: The
transformations A.3.1 require that ý´ = dy´/dx´ = mL/M –1(dy/dx ) = mL/M–1 ý.) The function v = ýx–L/M+1 is a first differential invariant, and it can be shown that the
most general first differential invariant is a function of u and v.
For all three of the partial differential equations in Eq. (A.1.l), the principal differential equation for y(x) is of second order. Hence the use of u and v as new variables reduces the principal differential equation to first order. I call the first-order differential equation in u and v the associated differential equation. When the second-order principal differential equation is not readily solvable, the first-order associated differential equation may be studied geometrically by means of its direction field.
A.4. ASYMPTOTIC BEHAVlOR
Before we turn to a concrete example of how this procedure works, certain additional generalities need to be described. There is a solution that is invariant not
202 |
Appendix A |
just to one group of the family, but to all |
groups of the family, namely, |
c = AzL/Mt– N / M , where A is a constant that is determined by the structure of the partial differential equation. When the ratios L/M and N/M are both negative, this totally invariant solution obeys the partial boundary and initial conditions c(∞,t) = 0 and c(z,0) = 0. Often, technologically interesting solutions of the partial differential equation obey the same boundary and initial conditions. For those that do, the totally invariant solution, under additional conditions described below, gives their asymptotic behavior for large z (Dresner, 1993, “General Properties”).
The additional conditions are these: The solutions of the partial differential equation must be ordered according to their boundary condition at z = 0. This means
that if c1(0,t) ≥ c2(0,t) for t > 0, and if both c 1 |
and c2 obey the boundary and initial |
conditions c(∞,t) = 0 and c(z,0) = 0, then c1(z,t) |
≥ c2(z,t) for all z and t. The ordering |
condition is fulfilled for the partial differential equations (l), (2), and (3) given above. For, these partial differential equations have the general form of conserva-
tion equations, namely, S(c)ct + qz = 0, where S > 0, and where ∂q/∂cz |
≥ |
0; it can |
be shown that solutions of such conservation equations for which c(∞,t) = 0 and c(z,0) = 0 are ordered according to their boundary condition c(0,t) (Dresner, 1993, “General Properties”); the argument is quite similar to that given in Section B.2 of Appendix B.
A.5. EXAMPLE: THE SUPERFLUID DlFFUSlON EQUATION
To show how the procedure outlined above works let us consider partial
—
differential equation (3), for which M = 2, N = –3, L = –4, and A = 4/3√3. The principal ordinary differential equation is
bd(ý1/3)/dx + xý – ay = 0 |
(A.5.1) |
Choosing as an invariant and a first differential invariant p = u1/2 = xy1/2 and q
– v1/3 = xý1/3, we find an associated differential equation
dq/dp = 2p(bq – q3 + ap2)/b(2p2 + q3) |
(A.5.2) |
Different choices of a and b correspond to different physical problems, i.e., to different boundary and initial conditions. Three problems of technological interest
correspond to the following boundary and initial conditions: (1) c(0,t) = 1, c(∞,t) = 0 and c(z,0) = 0 (α= 0, b = 4/3); (2) cz(0,t) = –1, c(∞,t) = 0 and c(z,0) = 0 (a = 1,
b = 2); (3) c dz = 1, c(∞,t) = 0 and c(z,0) = 0 (a = –1, b = 2/3). Since the partial differential equation describes heat transport, we can use the language of that subject to describe these problems: (1) is the problem of an initially uniform semi-infinite tube of He-II the temperature of whose front face is raised and clamped at t = 0; hence, it is called the clamped-temperature problem; (2) is the clamped-flux problem for the same semi-infinite tube; and (3) is the problem of a
The Method of Similarity Solutions |
203 |
tube infinite in both directions subjected to a sudden heat pulse per unit area at the plane z = 0 at time t = 0; it is called the pulsed-source problem.
Different values of a and b lead to different forms of the principal and associated ordinary differential Eqs. (A.5.1) and (A.5.2). For the clamped-tempera- ture problem (1) and the pulsed-source problem (3), the principal differential equation is analytically solvable:
(1) |
c(z,t) = 1 |
— |
(A.5.3a) |
–x/(x2 + a2)1/2, a2 = 8/3√3, x = z/t 3/4 |
|||
(3) |
|
— |
|
c(z,t) = t– 3/2(4/3√ 3)/(x4 + b4)1/2 , |
|
||
|
b = 2[G(1/4)]2/3(3p)1/2 = 2.854535, x = z/t 3/2 |
(A.5.3b) |
|
Both of |
the solutions (A.5.3a) and (A.5.3b) are asymptotic to the totally invariant |
||
|
|
— |
|
solution c(z,t) = (4/3√ 3)z–2t3/2 when x >> 1, as expected. Solutions for different |
|||
values of c(0,t) or |
c dz than those given above can be obtained |
from the |
|
solutions (A.5.3a) and (A.5.3b) by scaling c(z,t) with the group (A.1.2) or y(x) with the associated group (A.3.1). Scaling does not affect the asymptotic limit of the solutions since it is totally invariant to the family (A.1.2).
For the clamped-flux problem (2), there is no simple solution to the principal differential equation, which must be solved numerically subject to the two-point boundary conditions ý(0) = –1 and y(∞) = 0. To avoid the labor of the shooting method, we turn for help to the associated differential equation.
Corresponding to the solution y(x) we seek, there is a curve in the (p,q) plane which we now must identify. Shown in Fig. A.1 is the fourth quadrant of the direction field of (A.5.2) for a = 1 and b = 2. Only the fourth quadrant interests us since p > 0 and q < 0 (because y > 0 and ý < 0). The curves C1 and C2, the loci of zero and infinite slope dq/dp, respectively, divide the direction field into regions in
which the slope dq/dp has one sign only. The intersections of these curves, the points O (0,0) and P (2/33/4, –2/31/2), are the singular points of (A.5.2).
The totally invariant solution c = AzL/Mt –N/M corresponds to the solution y = AxL/M of the principal differential equation, which is invariant to the associated group (A.3.1). For this solution, the invariant u = A and the first differential invariant
v = (L/M)A. Thus the totally invariant solution maps into a single point in the (p,q) plane, namely, the point (A1/2,[(L/M)A]1/3), which is the singular point P. (That the
totally invariant solution always maps into a singular point in the (u,v) plane follows from the fact that for the solution y = AxL/M of the principal differential equation, du = dv = 0 as x changes.) Thus the curve in the (p,q) plane that corresponds to the solution y(x) that we seek must pass through the singular point P. Furthermore, since P corresponds to the asymptotic behavior y ~ AxL/M of the solution y(x) that we seek, it corresponds to the limit x = ∞. When x = 0, on the other hand, p = q = 0, and the curve in the (p,q) plane must also pass through the origin O. Only the
204 |
Appendix A |
Figure A.1. The fourth quadrant of the direction field of Eq. (A.5.2) when a = 1 and b = 2.
separatrix S does so. It is the curve in the (p,q) plane defined by the solution y(x) that we seek.
Near the origin in the (p,q) plane, the integral curves behave linearly, i.e., p = –Bq. Substituting the definitions of p and q, we find that [y(0)]1/2 = –Bý1/3(0), which means that y(0) = B2 since ý(0) = –1. To find the value of B, we proceed as follows:
Since the point P is a saddle point, two separatrices cross it. We can find their slopes |
|
by applying L’Hospital’s rule to |
— |
(A.5.2.) The negative slope m = –31/4(3 + √17)/6 |
|
= –1 .562422. Using this slope to get starting values p = pp – e, q = qp – me near P, we can integrate (6) numerically from P to O and find B = 0.912582.
Now we have values of both y(0) (= B2) and ý(0) (= –1), so we can integrate (A.5.1) in the forward direction. Here a slight problem arises because integrating (A.5.1) in the forward direction carries us along the separatrix S from O towards P. Because the integral curves in the (p,q) plane diverge away from O, integration in the direction from O to P is unstable: a small error (roundoff or truncation) throws us off the separatrix S and we eventually diverge to one side or the other. This instability is reflected in a corresponding instability as we attempt to integrate (A.5.1) in the forward direction. Nevertheless, as a practical matter, it is possible to advance to about x ~ 1 without undue errors; the computed behavior can then be joined to the known asymptotic behavior to achieve a reasonable estimate of y(x).
The Method of Similarity Solutions |
205 |
Fortunately, a way exists to integrate (A.5.1) in the backward, stable direction. We proceed as follows: (1) we choose a point (p,q) on S close to P; (2) guess a (large) value of x, say x1; (3) calculate y1 and ý1 from the chosen values of p and q ; and (4) use these values of as starting values for a backward, stable integration from
x1 to 0. This procedure works for the following reason. Any image point of x1, y1, ý1, say mx1,m-2y1, m–3ý1, has the same values of p and q as the point x1, y1, ý1 itself
because p and q are functions of the group invariants u and v. Thus any value of x can be made to correspond to anyp and q on the separatrix. In general, the backward integration will not give the curve for which ý(0) has some specified value. But once the curve y(x) has been calculated, it can be scaled with the associated group to a curve with any desired ý(0).
A.6. INFORMATION OBTAINABLE BY GROUP ANALYSIS ALONE
Since all the curves y(x) corresponding to different values of ý(0) are images of one another under the associated group (A.3.1), all have the same value B of –y1/2(0)/ý1/3(0) because this quantity is invariant to transformations of the associated group. (Note that the point x = 0 transforms into the point x´ = 0.) From this it
immediately follows that c(0,t) = B2ý2/3(0)t1/2. This formula gives the dependence of the temperature of the front face c(0,t) on the time t and the clamped flux –ý1/3(0),
which are the only two parameters in the problem on which it can depend. We could have obtained this formula directly from knowledge of the associated group so that by group analysis alone we can obtain a formula for c(0,t) correct up to a single undetermined constant. To find the value of the constant, however, we must integrate the associated differential equation.
The method outlined here does not depend on the partial differential equation being linear. On the other hand, it does depend on the partial differential equation being invariant to a one-parameter family of one-parameter stretching groups. This is a high degree of algebraic symmetry that is only found in the simplest equations. But many equations of technological interest have the high symmetry required and so can be dealt with by the method of this Appendix.
This page intentionally left blank.
Appendix B
Stability of the MPZ
B.1. THE ORDERING THEOREM
The task in this appendix is to show that the MPZ is unstable and that it separates quenching initial conditions from those that recover. To do this we shall study some general properties of the solutions of the time-dependent partial differential Eq. (4.4.1). To simplify the notation and bring it into conformity with the notation of Section A.4, let us set
(B.1.1)
and then replace S/k by S(c) and QP/A – qP/A by Q(c). Then Eq. (4.4.1) takes the form
S(c)ct = czz + Q(c) |
(B.1.2) |
The first result we prove is the following ordering theorem: If c1(z,t) and c2(z,t) are two solutions of the partial differential Eq. (B.1.2) for which c1(a,t) = c2(a,t) and
c1(b,t) = c2(b,t) but for which c1(z,0)≥ c2(z,0), a ≥ z ≥ b, then c1(z,t) ≥ c2(z,t), a ≥ z ≥ b, for all t ≥ 0. In other words, if two solutions obey the same boundary conditions
but one starts out bigger than the other, it always remains bigger.
B.2. PROOF OF THE ORDERING THEOREM
The proof of the ordering theorem is a straightforward application of the methods of Protter and Weinberger (1967). Let us subtract Eq. (B.1.2) written for c2 from Eq. (B.1.2) written for c1. If we set h = c1 – c2, we find
207
208 |
Appendix B |
Figure B.1. An auxiliary diagram to aid in the proof of the ordering theorem.
S(c)ht = hzz + [Q´(c) – S´(c)ct]h |
(B.2.1) |
In obtaining Eq. (B.2.1) we have used the law of the mean in the following two ways:
= (∂/∂t )[S(c)h], c2
and
Q(c1) – Q(c2) = Q´(c)η , c2 ≥ c ≥ c1
≥
c
≥
c1
(B.2.2)
(B.2.3)
Owing to this usage, we do not know the exact values of S(c) and Q´(c) – S´(c)ct in Eq. (B.2.1), but all we need to know is that S(c) > 0 and Q´(c) – S´(c)ct is bounded.
In terms of h, the boundary and initial conditions become h(a,t) = h(b,t) = 0 and h(z,0) >0, a ≥ z ≥b.What we need to prove is that h(z,t) ≥ 0, a ≥ z ≥ b, for all
t > 0. To do this we employ a trick of Protter and Weinberger (1967) to overcome the fact that we do not know the algebraic sign of Q´(c) – S´(c)ct. We introduce the auxiliary variable z = he–lt , which satisfies the following partial differential equation and boundary and initial conditions (cf. Fig. B.1):
S(c)zt = zzz + [Q´ (c) – S´(c)ct –lS(c)]z
z(a,t) = z(b,t) = 0, z(z,0) ≥ 0, a ≥ z ≥ b
(B.2.4)
(B.2.5)
Stability of the MPZ |
209 |
If z has a minimum in the rectangle ABCD or on its boundary, that minimum must be ≥ 0. Now we prove that z cannot have a negative minimum at a point P in
the interior of the rectangle ABCD in Fig. B.1. For at such a minimum, zt(P) = 0, |
|
zzz (P) ≥ 0, and z(P) < 0. If we choose |
|
l > max[Q´(c) – S´(c)ct]/min[S(c)] |
(B.2.6) |
we find that the right-hand side of Eq. (B.2.4) is > 0 while the left-hand side = 0, a contradiction. Furthermore, z cannot have a negative minimum at a point Q in the interior of line segment BC. For at such a minimum, zzz(Q) ≥ 0 and z(Q) < 0 so that zt (Q) > 0. Then yet smaller values of z would exist in rectangle ABCD just below point Q, again a contradiction.
Thus the minimum value of z must lie on segments AB, AD, or DC of the boundary. The minimum value there is 0, so that z ≥ 0 in or on the boundary of rectangle ABCD, which is what we wished to prove.
B.3. APPLICATION OF THE ORDERING THEOREM TO THE MPZ
The immediate impact of this theorem is that solutions whose initial state is everywhere (a = –∞, b = ∞) greater than the MPZ continue greater than the MPZ and solutions whose initial state is everywhere less than the MPZ continue less than the MPZ. Now the solutions of partial differential Eq. (B.1.2) can evolve in one of three ways: they can grow without bound in amplitude or extent (quench), or they
can approach one of the two steady states c = 0 or cMPZ. If is cMPZ unstable against perturbation, it is commonly assumed that the initial conditions that occur in
practice diverge away from cMPZ. The initial states < cMPZ and > 0 must recover (i.e., approach the steady state c = 0) whereas initial conditions > cMPZ must quench.
In point of fact, however, all instability of a steady state means is that in every neighborhood of the steady state an initial condition exists whose corresponding solution diverges from the steady state. Stability, on the other hand, is a stronger statement. It means that there is some neighborhood of the steady state in which the solution corresponding to every initial condition approaches the steady state. In the case of an unstable steady state, there may well be some initial conditions that approach the steady state, but not all of them. As noted above, it is commonly assumed that the initial conditions that occur in practice diverge from the steady state.
8.4.LAGRANGIAN FORMULATION AND THE STABILITY OF STEADY STATES
To prove the instability of the MPZ, we note that Eq. (B.1.2) can be written as an Euler–Lagrange equation