No abstract

In this paper, we study the following water wave model with a nonlocal viscous term: u_{t}+u_{x}+\beta u_{xxx}+\frac{\sqrt{\nu}}{\sqrt{\pi}}\frac{\partial}{% \partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{t-s}}\,ds+uu_{x}=\nu u_{xx}, where {\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{% t-s}}\,ds} is the Riemann–Liouville half-order derivative. We prove the well-posedness of this model using diffusive realization of the half-order derivative, and we discuss the asymptotic convergence of the solution. Also, we compare our mathematical results with those given in [5] and [14] for similar equations.

Communicated by: R. RODRIGUEZ Funding information PHC Utique ASEO MSC Classification: 35Q35; 35Q53In this paper, we numerically study the water wave model with a nonlocal viscous termds is the Riemann-Liouville half-order derivative in time. We propose and compare different numerical schemes based on the diffusive realizations of fractional operators.

In this paper, we numerically investigate the BBM‐Burgers equation with a nonlocal viscous term u t + u x − β u t x x + ν π ∂ ∂ t ∫ 0 t u false( s false) t − s d s + γ u u x = α u x x , where 1 π ∂ ∂ t ∫ 0 t u false( s false) t − s d s is the Riemann‐Liouville half derivative. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in for similar models.

We consider here semilinear Schrödinger equations with a non standard dispersion that is discontinuous at x = 0. We first establish the existence and uniqueness of standing wave solutions for these equations. We then study the orbital stability of these standing wave into a subspace of the energy space that where classical methods as the concentration-compactness method of P.L. Lions can be used.