Similarity solutions of nonlinear third-order dispersive PDEs: The first critical exponent

This study investigates the behaviour of blow-up and global similarity solutions for the nonlinear dispersion equation (NDE), ut=(|u|nu)xxx±(|u|p−1u)xxinR×R+,n>0andp>n+1,and attempts to give some aspects analytically and numerically. One can easily see that the proposed NDE is more complicated and hence more difficult than the corresponding semilinear equation. We will particularly pay attention to the first critical exponent p=p0=n+2, which helps us to simplify the rescaled nonlinear equation and compare with semilinear ones studied in the literature.