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By Eschenburg J. H., Tribuzy R.

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33) where "R is an arbitrary rectangle in the (x, t) plane, and (j> is an arbitrary "test" function with continuous first derivatives in R and

Consider the F curve in Fig. 16. Two shocks are formed corresponding to the points of inflexion P and Q with families of equal area chords typified by PlP2 and <2,<22. As time goes on the points Qx and P2 approach each other until the stage in Fig. 166 is reached where a common chord cuts off lobes of equal area for both humps. At this stage the characteristics corresponding to P2 and Q[ are the same, and therefore the shocks have just combined into one as shown in the (x,t) diagram Fig. 17. All the characteristics between Q2 and P{ have now been absorbed by one or other of the shocks; a single shock proceeds using chords PXQ2 as in Fig.

However, the fact that the shock transition becomes very wide as (p 2 — Pi)/ p,—>0, for fixed v, means that in any problem where the shocks ultimately tend to zero strength as /-»oo, there may be some final stage with extremely weak shocks when the discontinuous theory will be invalid. This is often a very uninteresting stage, since the shocks must be very weak. Otherwise, we can say that the two alternative ways of improving on the unacceptable multivalued solutions agree. The use of discontinuous shocks is the easier analytically and can be carried further in more complicated problems.

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(1,1)-geodesic maps into grassmann manifolds by Eschenburg J. H., Tribuzy R.

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