Consider the hyperbolic nonlinear Schrödinger equation $\mathrm {(HNLS)}$ over $\mathbb R^d$ $$ iu_t + u_{xx} - \Delta_{\textbf{y}} u + \lambda |u|^\sigma u=0. $$ We deduce the conservation laws associated with $\mathrm {(HNLS)}$ and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including hyperbolically symmetric solutions, spatial plane waves and spatial plane waves, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a stability result for both spatial plane waves and spatial standing waves with respect to small $H^1$ perturbations.

In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of electron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that $\|u_0\|_{\dot{H}^{\frac12}}+\|B_0\|_{\dot{H}^{\frac12}} +\|\nabla B_0\|_{\dot{H}^{\frac12}}$ is sufficiently small. Moreover, by applying a cut-off function and generalized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time $t\rightarrow\infty$. Finally, the algebraic decay rate of the weak solution of electron inertia Hall-MHD system is established by using Fourier splitting method and the properties of decay character.

We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $\epsilon$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0 with $\epsilon$. We construct outer solutions for time located in complex sectors that are kept away from the origin at a distance equivalent to a positive power of $|\epsilon|$ and we build up a related family of sectorial holomorphic inner solutions for small time inside some boundary layer. We show that both outer and inner solutions have Gevrey asymptotic expansions as $\epsilon$ tends to 0 on appropriate sets of sectors that cover a neighborhood of the origin in $ \mathbb{C}^{\ast}$. We observe that their Gevrey orders are distinct in general.