Methods in Nonlinear Plasma Theory by Ronald C. Davidson

By Ronald C. Davidson

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Phys. JETP 10, 570 (1960). G . K a l m a n , Ann. Phys. (New York) 10, 1, 29 (1960). L . T o n k s a n d I. L a n g m u i r , Phys. Rev. 3 3 , 195 (1929). P . B e r t r a n d a n d M . R. F e i x , Phys. Lett. A 2 8 , 68 (1968). R . C . D a v i d s o n a n d P . P . S c h r a m , Nucl. Fusion 8 , 183 (1968). R. Courant and K. O. Friederichs, "Supersonic Flow and Shock W a v e s . " Wiley ( I n t e r s c i e n c e ) , N e w Y o r k , 1948. T . M . O ' N e i l , Phys. Fluids 8 , 2255 (1965). E . A .

4 43 EXTENSIONS OF THE MODEL and ^3(^0 » ) T ^3(^0 = (51) » 0). T h i s gives a complete description of t h e p r o b l e m in L a g r a n g i a n variables. T h e explicit b e h a v i o r in Eulerian variables d e p e n d s on t h e details of t h e initial conditions chosen for t h e p r o b l e m . However, it is clear that coherent oscillations at t h e u p p e r h y b r i d frequency are m a i n t a i n e d for all t i m e in t h e region of initial excitation. I t is readily verified that t h e transformation from L a g r a n g i a n to Eulerian variables r e m a i n s u n i q u e provided Ο > - 2 ^2ω\W (52) ^ pe and 3 2 V(x ) dx n pe ω < 0 fleV/2 ρ θ υ (53) 2 pe Inequalities (52) a n d (53) r e d u c e to (22) a n d (23) for Q —> 0.

8. 9. 10. 11. 12 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. D . J. K o r t e w e g a n d G . d e V r i e s , Phil. Mag. 39, 4 2 2 (1895). C . S. G a r d n e r a n d G . K . M o r i k a w a , R e p . N Y U - 9 0 8 2 . C o u r a n t I n s t , of M a t h . , N e w Y o r k U n i v . , N e w Y o r k , 1960. M . D . K r u s k a l a n d N . J. Z a b u s k y , A n n u . R e p . M A T T Q - 2 1 . P l a s m a P h y s . L a b . , P r i n c e t o n U n i v . , P r i n c e t o n , N e w J e r s e y , 1963. N .

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