A generalization of integrable peakon equations with cubic nonlinearity and the Degasperis-Procesi equation with peakon solutions is proposed, which is associated with a 3×3 matrix spectral problem with two potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The generalization is exactly a negative flow in the hierarchy and admits exact solutions with N -peakons and an infinite sequence of conserved quantities. Moreover, a reduction of this hierarchy and its Hamiltonian structures are discussed.

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In this paper, we give the solutions on a periodic background in terms of the determinant form for the derivative nonlinear Schrödinger equation. Because its rogue wave on a periodic background has been studied, we investigate only the breather and breather-rogue wave on a periodic background for the derivative nonlinear Schrödinger equation. We obtain Kuznetsov–Ma breather, Akhmediev breather and spatio-temporal breather on a periodic background for this equation. In addition, we mainly focus on three types of the breather-rogue wave on a periodic background: (1) the interaction between a Peregrine soliton and a breather; (2) the interaction between a Peregrine soliton and two breathers; (3) the interaction between a second-order rogue wave and a breather. For the first type, we analyse the effects of the free parameters on its dynamical behaviour. The second type is described as ‘rogue wave quanta’ on a periodic background. The third type has two spatial-temporal distribution structures: the fundamental structure and the triangular structure.

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A hierarchy of new nonlinear evolution equations which contains the modified Korteweg–de Vries equation is proposed. With the aid of the inverse scattering transformation, N-soliton solutions of the first two nonlinear evolution equations in this hierarchy are derived. Based on the theory of algebraic curve, the corresponding flows are straightened under the Abel–Jacobi coordinates. The meromorphic function ϕ and the hyperelliptic curve Kn are introduced by which quasiperiodic solutions of the first two nonlinear evolution equations are constructed according to the asymptotic properties and the algebrogeometric characters of ϕ and Kn.