Exact solutions of the nonlocal Sawada–Kotera equation in the Alice–Bob system

To find the group invariant solution, a new shifted parity and delayed time reversal symmetries are used in order to construct Alice–Bob systems. With the help of simple assumptions and of the [Formula: see text] symmetry, the intrinsic two-place model of the Sawada–Kotera (SK) system is elucidated. A new form of the [Formula: see text]-soliton solution for the nonlocal SK equation is obtained, and dynamic properties of the [Formula: see text]-soliton solutions with different values are discussed, respectively. In addition, the breather solution for the AB–SK system is also explicitly identified.

Peakon Solutions of Alice-Bob b-Family Equation and Novikov Equation

By requiring B=P^sT^dA and substituting u=A+B into the b-family equation and Novikov equation, we can obtain Alice-Bob peakon systems, where P^s and T^d are the arbitrary shifted parity transformation and delayed time reversal transformation, respectively. The nonlocal integrable Camassa-Holm equation and Degasperis-Procesi equation can be derived from the Alice-Bob b-family equations by choosing different parameters. Some new types of interesting solutions are solved including explicit one-peakons, two-peakons, and N-peakons solutions.