Another possible model equation for long waves in nonlinear dispersive systems

In the same spirit in which Benjamin, Bona, and Mahoney modified the Korteweg-de Vries equation (U_x+U_t+UU_x+U_xxx=0) to obtain the so-called BBM equation, U_x+U_t+UU_x?U_xxt=0, we propose a different modification: U_x+U_t+UU_x+U_xtt=0. The advantages in this equation are 1) the system is conservative since it can be derived from the Lagrangian density $\text{L=}\text{1}\text{2}\text{θ}{}_{\mathrm{x}}\text{θ}{}_{\mathrm{t}}\text{+}\text{1}\text{2}\text{θ}{}^2{}_{\mathrm{x}}\text{+}\text{1}\text{6}\text{θ}{}^3{}_{\mathrm{x}}\text{?}\text{1}\text{2}\text{θ}{}^2{}_{\mathrm{xt}}\text{,}\text{where}\text{θ}{}_{\mathrm{x}}\text{≡ U;2)}$ for large wave-numbers |k|, the infinitesimal-wave phase speed falls off like $\text{1}\text{|k|}$, in accord with physical intuition; 3) since the equation is of second order in t, both U and U_t can be independently specified for t = 0. Several conservation laws satisfied by solutions to this equation are given.