Peaked and Smooth Solitons for K*(4, 1) Equation

This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com ". Abstract: The qualitative theory of differential equations is applied to K * (3, 1) equation, u t = u x u ? 2a(uu xxx ? 2u x u xx). Our procedure shows that K * (3, 1) equation has the regular peakon soliton, cuspon soliton and smooth soliton solutions when sitting on non-zero constant pedestal lim x!AE1 u ¼ A 6 ¼ 0; or possesses compacton solutions only when lim x!AE1 u ¼ A ¼ 0: In particular, mathematical analysis and numerical graph are provided for those peakon, cuspon, compacton and smooth soliton solutions.