N 1. (q k+1 q k ) 2 + α 3. k=0


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1 Teoretisk Fysik Handin problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of nonlinear oscillators to see how the energy distribution is randomized in a system with many degrees of freedom. Instead of the rapid transition to a uniform energy distribution one would naively expect, they observed an extremely slow randomization. In subsequent work by Zabusky and Kruskal 2 the relation of the FermiPastaUlam model to the Kortewegde Vries (KdV) equation was noted, and in a numeric solution of the KdV equation the appearance of solitary wave pulses was observed which they named solitons. This work motivated the search for a analytic solution of the KdV equation which was achieved two years later. 3 This latter work started a period of very intensive research on a class of nonlinear partial differential equations which today are known as soliton equations. The FermiPastaUlam model describes a weakly nonlinear, fixedend onedimensional chain of (N 1) moving mass points and is given by the Hamiltonian H = k=1 1 2m p2 k + K 2 k=0 (q k+1 q k ) 2 + α 3 k=0 (q k+1 q k ) 3 (1) where q 0 q N 0 and q k and p k are the coordinate and momentum for the kth particle, and the parameter α is a small nonlinear coupling; the positive parameters m and K are the particle mass and spring constant, respectively. As was realized by Zabusky and Kruskal (ZK), the equations of motion following from this Hamiltonian (the dots mean differentiation with respect to time t), m q k = K(q k+1 2q k + q k 1 ) + α[(q k+1 q k ) 2 (q k q k 1 ) 2 ] (2) in the lowestorder continuum limit take the following form, q tt = (c ɛq x )q xx (3) with some constants c 0 and ɛ. Eq. (3) can be viewed as ordinary wave equation whose wave speed c depends on the spatial derivative q x, c 2 = (c ɛq x). Typical solutions of this equations for positive ɛ describe pulses moving to the left or right while changing shape and steepening until their leading edge develops a vertical shock front (like water waves in the sea before they break close to the beach), at which point Eq. (3) looses validity. Nonetheless, prior to the formation of the shock, it was found that Eq. (3) provides a quite reasonable description of the Fermi PastaUlam model behavior. ZK found that, to avoid shock wave behavior, one has to take into account dissipation, which can be done by including the nextorder spatial derivative term, q tt = c 2 0q xx + ɛq x q xx + βq xxxx (4) for some constant β. For both convenience and simplicity, ZK now insist on periodic boundary conditions, restrict their attention to waves traveling on one direction only, and elect a moving frame. After replacing x by x = x c 1 t, t by t = c 2 t, q x (x, t) by c 3 u( x, t) in Eq. (4) with particular constants c j and neglecting terms proportional to ɛ 2 they obtained the KdV equation in the following form, u t + uu x + δ 2 u x x x = 0, with δ a (small) constant. In the following we will write this equation as u t + uu x + δ 2 u xxx = 0, u = u(x, t) (5) 1 Los Alamos Scientific Report, unpublished 2 Zabusky and Kruskal, Phys. Rev. Lett. 15, 240 (1965) 3 Gardner, Greene, Kruskal and Miura, Phys. Rev. Lett. 19, 1095 (1967) 1
2 with u = u(x, t), to simplify notation. In their short paper referred to above, ZK first recalled the welllocalized traveling wave solutions of the the KdV equation, with U(x) = u(x, t) = U(x ct) (6) 3c cosh 2 ( c(x x 0 )/2δ), (7) which they describe as solitary wave pulses. The amplitude of such a pulse is proportional to its velocity c > 0, and the constant x 0 equal to the location of its centerofmass at t = 0. ZK then discuss the equation following from (3) without dissipative term, together with the initial condition Using the fact that u t + uu x = 0, (8) u(0, x) = cos πx. (9) u(t, x) = f(x u(t, x)t) (10) is an implicit solution of (8) for any differentiable function f, they find that the solution of Eqs. (8) and (9) tends to become discontinuous at x = 1/2 and t = T B = 1/π. They then describe their results of a numeric solution of the KdV equation with small but nonzero dissipative term, δ = (11) They find that for times smaller than the breakdown time T B, the solution is nearly the same as for δ = 0, but then the dissipative term becomes important. For nonzero δ there is no breakdown any more, but instead solitary wave pulses appear: after a while the solution looks very much like a superposition of special solutions in Eqs. (6) and (7), with different values of c and x 0, 4 and the localized pulses are very stable, preserving shape and speed always except when two of them meet, in which case there is a certain interaction period in which the pulses merge. After such interactions the individual pulses reappear which the same shapes and velocities as before the interaction. ZK also observe that, during the interaction period, the joint amplitude of the interacting solitons decreases, in contradistinction to what would happen if the pulses overlapped linearly. In this problem you are asked to fill in the details of the derivations outlined above (part A). You are then asked to solve the KdV equation numerically and, in particular, reproduce the ZK result (part B). More specific instructions together with further hints will be given below. 4 Note that this is very remarkable since the KdV equation is nonlinear, and the superposition principle therefore does not hold. 2
3 Part A: To do: Derive Eq. (2) from Eq. (1). Hint: Use the following variational principle (= Hamiltonian principle), ( t2 N 1 ) δ dt p k (t) q k (t) H(p 1 (t),..., p N 1 (t), q 1 (t),..., q N 1 (t)) t 1 k=1 = 0. (12) From this obtain 2(N 1) first order differential equations of the form q k =..., ṗ k =... (= Hamilton equations), which, after eliminating half of the variables, yield (2). Derive in detail the continuum limit of Eq. (2) as described above. Hint: To obtain the continuum limit in Eq. (3), introduce x = ka where a > 0 is the lattice constant (i.e. the distance of adjacent mass points when the chain is not moving), and write q(t, x) = q k (t) where x = ka. Approximate q k±1 (t) = q(t, x ± a) by a Taylor series, keeping only certain lower order derivative terms. Do not forget to give the relation between the parameters (i.e. write down the formulas for the parameters c 0 and ɛ in Eq. (3) in terms of m, K, a, α, etc.). Generalize the argument to get Eq. (4). Give then all the details leading to Eq. (5). Verify in detail the shock wave behavior of Eq. (8), as described above. Hint: Verify that Eq. (10) is an implicit solution of Eq. (8). Use (10) to sketch the solution of Eqs. (8) and (9) for a few different times between 0 and T B = 1/π, to show the steepening of the wave. Verify that the wave breaks at the time T B. (If you cannot show this analytically, you can also hand in numeric verifications of these facts produced with the help of MATLAB.) Derive the solutions in Eqs. (6) and (7) from the KdV equation (5). Hint: This is easy if you read Chapter 9 in FMM, of course. Part (B): To do: Solve Eq. (5) for δ = numerically by using the finite difference method (FDM; see Section 8.1 in FMM), for 0 x 2 and periodic boundary conditions, u(x + 2, t) = u(x, t). Test your program by comparing the numeric solution with the analytical solution given in Eqs. (6) and (7) for x 0 = 0.4 and some value of c for times 0 t 0.6/c. Hand in a plot showing your result together with a printout of your MATLAB program. Hint: We suggest you use the following discretization of the KdV equation, with u i,j+1 = u i,j k 6h (u i+1,j + u i,j + u i 1,j ) (u i+1,j u i 1,j ) δ 2 k 2h 3 (u i+2,j 2u i+1,j + 2u i 1,j u i 2,j ) (13) x i = ih, h = 2, i = 0, 1, 2... N 1 N t j = jk, k = t max, j = 0, 1, 2... M (14) M 3
4 and u i+n,j = u i,j (periodic boundary conditions). The initial condition is u i,0 = u(x i, 0), of course. Show that Eq. (13) indeed is a discretization of Eq. (5). To solve these equations on a computer, we suggest you use MATLAB (you can also use other program tools if you prefer). A similar MATLAB program which you can use as a starting point can be found on the course homepage. You will find that not all values of c, h and k are numerically stable, and you need to experiment to find good values. Values that are numerically stable are e.g. c = 0.1, h = 0.06, k = 0.005, (15) but you can get better results with smaller values for h and k. 5 program by either decreasing the ratio k/h 3 or/and decreasing c. We could stabilize our Find a simpler FDM discretization of the KdV equation. Implement your discretization in MATLAB (e.g.) and test your program. Hand in your program and one representative plot showing your test results. 6 Hand in plots showing your numerical solution of the KdV equation with the following initial conditions, (i) u(x, 0) = A cos(πx) 0 t 2/(Aπ) (ii) u(x, 0) = 3U(x) for x 0 = 0.4, 0 t 0.6/c (iii) u(x, 0) = U(x) for x 0 = 0.4, 0 t 0.6/c (16) with U(x) in Eq. (7), and some value of c > 0 and A > 0 which you choose, showing the solution in a six equidistant times (including initial and final time) in the given time interval. Discuss your results. Hint: ZK give the result for (i) with A = 1 which therefore would be the preferred choice, but to get reasonable results for that you need much smaller h and k values than the ones given above. If you computer cannot handle that choose a smaller values for A like A = 0.1. The initial condition u(x, 0) = 1 2n(n + 1)U(x) for n = 1, 2,... is known to be a nsoliton solution at the moment when all solitons are at the same spot (nsoliton collision). For t > 0 the solitons move to the right and separate. For negative initial condition one does not have any solitons but only dispersive waves moving to the left. Try our different initial conditions and different values for k and h in your program. Hand in one plot (iv) with an initial condition of your own choice which you find interesting. Hint: It is fine if you choose u(x, 0) = AU(x) for some other parameters of c, x 0 and A > 0, but perhaps you find something more interesting. Results from an unstable program are not interesting, of course. 5 If your computer is slow and cannot handle such small values of h and k it is OK to have larger values. 6 A negative result (program based on your discretization cannot be made stable) is OK if described properly. 4
5 GENERAL INSTRUCTIONS: A MAIN GOAL IN THIS PROJECT IS TO WRITE A GOOD SCIENTIFIC REPORT. Your solution should be selfcontained and preferably hand written (except for the program code, of course). Give a clear description of your reasoning. Your plots should have a figure caption explaining what they display, and the axes should be labeled. (The plot produced by the MATLAB program provided by us is not OK as it comes out of the program! It is OK to write the labels and figure captions by hand.) Please put your papers together (no loose pages)! Collaborations are allowed but only up to two students in one team, and each of the students should hand in his/her own report. If you collaborate and/or hand in a report in LaTex or the like you should be prepared to be asked to explain your solution in an oral presentation. Do not forget to write name, ADDRESS and personal number on your papers. 5
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