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In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal's asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185]. [ABSTRACT FROM AUTHOR]

Abstract: In the literature, on the numerical solution of nonlinear time dependent partial differential equations, much attention has been paid to numerical processes which have the favourable property of being total variation bounded (TVB). A popular approach to guaranteeing the TVB property consists in demanding that the process has the stronger property of being total variation diminishing (TVD). For Runge–Kutta methods—applied to semi-discrete approximations of partial differential equations—conditions on the time step were established which guarantee the TVD property; see, e.g., [J. Comput. Phys. 77 (1988) 439; Math. Comp. 67 (1998) 73; SIAM Rev. 43 (2001) 89; SIAM J. Numer. Anal. (2002), in press; Higueras, Tech. Report, Universidad Pública de Navarra, 2002; SIAM J. Numer. Anal. 40 (2002) 469]. These conditions were derived under the assumption that the simple explicit Euler time stepping process is TVD. However, for various important semi-discrete approximations, the Euler process is TVB but not TVD—see, e.g., [Math. Comp. 49 (1987) 105; Math. Comp. 52 (1989) 411]. Accordingly, the above stepsize conditions for Runge–Kutta methods are not directly relevant to such approximations, and there is a need for stepsize restrictions with a wider range of applications. In this paper, we propose a general theory yielding stepsize restrictions which cover a larger class of semi-discrete approximations than covered thus far in the literature. In particular, our theory gives stepsize restrictions, for general Runge–Kutta methods, which guarantee total-variation-boundedness in situations where the Euler process is TVB but not TVD. [Copyright &y& Elsevier]