Exact travelling wave solutions for the modified Novikov equation ∗

which was discovered in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity. By using the perturbation symmetry approach [7], Novikov found the first few symmetries and a scalar Lax pair for Eq. (1), then proved that it is integrable [9]. Hone and Wang [5] gave a matrix Lax pair for the Novikov equation and found its infinitely many conserved quantities, as well as a bi-Hamiltonian structure. Then using the matrix Lax pair found by Hone and Wang [5], Hone, Lundmark and Szmigielski [4] obtained the explicit formulas for multipeakon solutions of Eq. (1). For other studies concerned with blow-up phenomenon, Cauchy problem of Eq. (1), we refer the reader to see [3, 6, 8, 10, 13, 14].