In this work, the Riemann–Hilbert (RH) problem is employed to study the multiple high-order pole solutions of the cubic Camassa–Holm (cCH) equation with the term characterizing the effect of linear dispersion under zero boundary conditions and nonzero boundary conditions. Under the reflectionless situation, we generalize the residue theorem and obtain the multiple high-order pole solutions of cCH equation by solving an algebraic system. During the process of establishing the solution of RH problem, to simplify the calculations involving the implicitly expressed of variables (x, t) in the solution, we introduce a new scale (y, t) to ensure the solution of RH problem is explicitly expressed with respect to it. Finally, the exact solutions are obtained for cases involving one high-order pole and N high-order poles.

The Kudryashov–Sinelshchikov–Olver equation describes pressure waves in liquids with gas bubbles taking into account heat transfer and viscosity. In this paper, we prove the existence of solutions of the Cauchy problem associated with this equation.

We first study the regularised version of a modified two-component Camassa–Holm shallow water system and obtain the energy estimates of the corresponding approximate solutions. Then, we present a sufficient condition which guarantees that these approximate solutions converge to a low regularity weak solution of the modified two-component Camassa–Holm shallow water system.

The well-posedness of the Ostrovsky equation is considered. Local well-posedness for data in $\tilde{H}^s(\mathbb{R})$ $(s\geq-\frac{1}{8})$ and global well-posedness for data in $\tilde{L}^{2}(\mathbb{R})$ are obtained.

We investigate the ‘clumping versus local finiteness' behavior in the infinite backward tree for a class of branching particle systems in ℝd with symmetric stable migration and critical ‘genuine multitype' branching. Under mild assumptions on the branching we establish, by analysing certain ergodic properties of the individual ancestral process, a critical dimension dc such that the (measure-valued) tree-top is almost surely locally finite if and only if d > dc. This result is used to obtain L1-norm asymptotics of a corresponding class of systems of non-linear partial differential equations.