Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

Global dissipative solutions of the Novikov equation

Pages: 1615 – 1633



Shouming Zhou (College of Mathematics Science, Chongqing Normal University, Chongqing, China)

Li Yang (College of Mathematics Science, Chongqing Normal University, Chongqing, China)

Chunlai Mu (College of Mathematics and Statistics, Chongqing University, Chongqing, China)


This paper is regarding the continuation of solutions to the Novikov equation beyond wave breaking. Our method is based on the characteristic of establishing new variables, then we transform the Novikov equation to a closed semilinear system on these new variables so that all singularities are resolved due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global dissipative solutions, which depends continuously on the initial data. Note that the nonlinearity of the Novikov equation is higher than the Camassa–Holm equation; this requires us to seek the high-order energy density and another conservative law.


Novikov equation, global dissipative solutions

2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

Received 12 February 2018

Accepted 9 May 2018

Published 7 February 2019