Studied here is the generalized Benjamin-Ono--Zakharov-Kuznetsov equation $$ u_t+u^pu_x+\alpha\mathscr{H}u_{xx}+\varepsilon u_{xyy}=0, \quad (x,y)\in \mathbb R ^2,\;\;t\in \mathbb R ^+, $$ in two space dimensions. Here, $\mathscr{H}$ is the Hilbert transform and subscripts denote partial differentiation. We classify when this equation possesses solitary-wave solutions in terms of the signs of the constants $\alpha$ and $\varepsilon$ appearing in the dispersive terms and the strength of the nonlinearity. Regularity and decay properties of these solitary wave are determined and their stability is studied.

The aim of this paper is to provide a proof of the (conditional) orbital stability of solitary waves solutions to the fractional Korteweg-de Vries equation (fKdV) and to the fractional Benjamin-Bona-Mahony (fBBM) equation in the $L^2$ subcritical case. We also discuss instability and its possible scenarios.

We prove existence and uniqueness of solutions for a system of PDEs which describes the growth of a sandpile in a silos with flat bottom under the action of a vertical, measure source. The tools we use are a discrete approximation of the source and the duality theory for optimal transport (or Monge-Kantorovich) problems.

We consider a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. Our main result shows the existence, uniqueness (up to translations) and local asymptotic stability of a traveling wave solution connecting two stable homogeneous steady states. In particular, we provide an extension to classical results on traveling wave solutions involving local diffusion. This extension to evolution equations with Riesz-Feller operators requires several technical steps. These steps are based upon an integral representation for Riesz-Feller operators, a comparison principle, regularity theory for space-fractional diffusion equations, and control of the far-field behavior.

We prove the existence of positive periodic solutions for the second order nonlinear equation $u'' + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

In this paper, we consider the Cauchy problem of the weakly dissipative Camassa-Holm equation. We prove the local well-posedness in the critical inhomogeneous Besov space $B^{3/2}_{2,1}( \mathbb R )$. This result depends on the apriori estimate of the nonlinear transport equation. Moreover, we show result for the finite time blowing up solution of the the weakly dissipative Camass-Holm equation which unifies previously known result. The proof relies on geometrical approach for the weakly dissipative Camass-Holm equation.