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In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point…
On the contrary to the common intuition, which suggests that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schroedinger equations, which include expulsive…
In this note we propose a new set of coordinates to study the hyperbolic or non-elliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically…
An inhomogeneous nonlinear Schr\"odinger equation is considered, that is invariant under $L^2$ scaling. The sharp condition for global existence of $H^1$ solutions is established, involving the $L^2$ norm of the ground state of the…
We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schr\"odinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one,…
We study the stability of traveling waves of nonlinear Schr\"odinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and…
We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such…
We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schr\"odinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an…
We determine conditions under which a generic gauge invariant nonautonomous and inhomogeneous nonlinear partial differential equation in the two-dimensional space-time continuum can be transform into standard autonomous forms. In addition…
We predict the existence of spatial-spectral vortex solitons in one-dimensional periodic waveguide arrays with quadratic nonlinear response. In such vortices the energy flow forms a closed loop through the simultaneous effects of phase…
In this paper, we present new results regarding the orbital stability of solitary standing waves for the general fourth-order Schr\"odinger equation with mixed dispersion. The existence of solitary waves can be determined both as minimizers…
We construct space quasi-periodic standing wave solutions to the nonlinear Schr\"odinger equations on R^d for arbitrary d. This is a type of quasi-periodic nonlinear Bloch-Floquet waves.
We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schroedinger equation (DNLS) with cubic nonlinearity, near the continuous limit.We construct a family of solutions…
In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a…
We revisit the topic of the existence and azimuthal modulational stability of solitary vortices (alias vortex solitons) in the two-dimensional (2D) cubic-quintic nonlinear Schr{\"o}dinger equation. We develop a semi-analytical approach,…
In his seminal work, Weinstein considered the question of the ground states for discrete Schr\"odinger equations with power law nonlinearities, posed on ${\mathbb Z}^d$. More specifically, he constructed the so-called normalized waves, by…
We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is…
We consider two-dimensional (2D) localized vortical modes in the three-wave system with the quadratic ($\chi ^{(2)}$) nonlinearity, alias nondegenerate second-harmonic-generating system, guided by the isotropic harmonic-oscillator (HO)…
Effects of nonlinear dynamics of solitary waves and wave modulations within the modular (also known as quadratically cubic) Korteweg - de Vries equation are studied analytically and numerically. Large wave events can occur in the course of…
This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and…