arXiv subjects

Mohamad Darwich

Publications and source records attributed to Mohamad Darwich.

11 recordsLinked to original sources

arXiv2024

Asymptotic stability of fast solitary waves to the Benjamin Equation

We prove the asymptotic stability of the high speed solitary waves to the Benjamin equation. This is done by establishing a Liouville property for the nonlinear evolution of the Benjamin equation around these solitary waves. To do this, inspired by Kenig-Martel-Robbiano 2011, we make use of the KdV limit of the Benjamin equation together with known rigidity property of the KdV flow. The main difficulties are linked to the presence of the Hilbert transform, that is a non-local operator, as well as the non-positivity of the quadratic part of the energy in the case $ γ<0$ which is the physical case.

math.AP↗

arXiv2024

On the Uniqueness and Orbital Stability of Slow and Fast Solitary Wave Solutions of the Benjamin Equation

This paper is devoted to the study of existence and properties of solitary waves of the Benjamin equation. The studied equation includes a parameter $γ$ in front of the Benjamin-Ono term. We show the existence, uniqueness, decay and orbital stability of solitary wave solutions obtained as a solution to a certain minimization problem, associated either with high speeds without a sign condition on the parameter $γ$ or with low speeds for the appropriate sign.

math.AP↗

arXiv2017

On the stability of the solitary waves to the rotation Benjamin-Ono equation

In this paper, we study several aspects of solitary wave solutions of the rotation Benjamin-Ono equation. By solving a minimization problem on the line, we construct a family of even travelling waves $ψ_{c,γ}$. We also study the strong convergence of this family and we establish the uniqueness of $ψ_{c,γ}$ for $γ$ small enough. Note that this improves the results in [5] where the stability of the set of ground states is proven.

math.AP↗

arXiv2017

On the generalized nonlinear Camassa-Holm equation

In this paper, a generalized nonlinear Camassa-Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method.

math.AP↗

arXiv2017

Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation

We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.

math.AP↗

arXiv2017

On the $L^{2}$-critical nonlinear Schrodinger equation with an inhomogeneous damping term

We consider the $L^2$-critical nonlinear Schrodinger equation with an inhomogeneous damping term. We prove that there exists an initial data such that the corresponding solution is global in $H^1(R^d)$ and we give the minimal time of the blow up for some initial data.

math.AP↗

arXiv2015

Some remarks on the nonlinear Schrödinger equation with fractional dissipation

We consider the Cauchy problem for the $L^2$-critical nonlinear Schrödinger equation with a fractional dissipation. According to the order of the fractional dissipation, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

math.AP↗

arXiv2013

On the invariant measures for the Ostrovsky equation

In this paper, we construct invariant measures for the Ostrovsky equation associated with conservation laws. On the other hand, we prove the local well- posedness of the initial value problem for the periodic Ostrovsky equation with initial data in $H^{s}(\mathbb{T})$ for $s>-1/2$.

math.AP↗

arXiv2013

On the $L^{2}$-critical nonlinear Schrödinger Equation with a nonlinear damping

We consider the Cauchy problem for the $L^{2}$-critical nonlinear Schrödinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

math.AP↗

arXiv2012

Blowup for the Damped $L^{2}$-Critical Nonlinear Schrödinger Equation

We consider the Cauchy problem for the $L^{2}$-critical damped nonlinear Schrödinger equation. We prove existence and stability of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

math.AP↗

arXiv2011

On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation

We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s > -1/2$, for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type spaces. This result is almost sharp if one requires the flow-map to be smooth.

math.AP↗