Time-dependent Solutions with Null Killing Spinor in M-theory and Superstrings

Imposing the condition that there should be a null Killing spinor with all the metrics and background field strengths being functions of the light-cone coordinates, we find general 1/2 BPS solutions in D=11 supergravity, and discuss several examples. In particular we show that the linear dilaton background is the most general supersymmetric solution without background with the additional requirement of flat space in the string frame. We also give the most general solutions for flat space in the string frame with RR or NS-NS backgrounds, and they are characterized by a single function