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Differential equation for X From (6), inserting into the other isocline equation yields at (E)+ xa, (g) =0 and X satisfies the second order differential equation xxxtt + (AX, - &)At, - AXtX, =0 which cannot be obtained from a Lagrangian L(Xt,X,,X). 20721 References 1. D. Lax, Ann. Math. Studies 33, 211 (1954); Comm. Pure Appl. Math. 10, 537 (1957); Selected Papers, 2 vol. (Springer-Verlag, New York, 2005). 2. D. Lax, J . Math. Phys. 5, 611 (1964). 3. G. Boillat, Nonlinear Hyperbolic Fields & Waves in: Recent Mathematical Methods in Nonlinear Wave Propagation ed.
Anal. 121, 131 (1992). 16. - J. Peng, Appl. Math. Lett. 11, 75 (1998). 17. -T. -J. Peng, Nonlinear Anal. 52,573 (2003). 18. G. Boillat and T. Z. (5), 15-A,197 (1978). 19. T. Ruggeri, Lett. Nuovo Cimento 22, 69 (1978). 20. T. Ruggeri and A. Strumia, Prog. Theor. Phys. 59, 2121 (1978). 21. G. Boillat and T. Ruggeri, Prog. Theor. Phys. 60,1928 (1978). Acknowledgments This paper was supported by MIUR PNN (Progetto di Ricerca di lnteresse Nazionale: Nonlinear Propagation and Stability in Thermodynamical Processes of Continuous Media.
The wave velocities follow from the equations (8) ( F - A1)Gv = 0 , 6~ + A ~ w= 0 . 27 It appears immediately that X = 0 is a solution (due to the reduction (8) to the first order of the differentiated system) which is incompatible with the constraints F ~+ w 6~ = 0 . Therefore p(w) and p'(w) satisfy the isocline equation (6). As an example consider the Monge-Amphre equation3 with constant coefficients H$tt Putting + 2K$tx + L$m + M + N($tt$xx $t = p , $z - $,",)= 0 . (9) = q we can write Pt + 2Kpx + Lq, + M - Np; H + Nqx =o qt - p x = 0 which is just of the form (7) with u = ( p , q ) .
CrossRoads vol 16 issue1 by CrossRoads
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