Nghiem V. Nguyen

Global existence for a coupled system of Schrödinger equations with power-type nonlinearities

In this manuscript, we consider the Cauchy problem for a Schrodinger system with power-type nonlinearitiesi∂∂tuj+▵uj+∑k=1majk|uk|p|uj|p−2uj=0,uj(x,0)=ψj0(x),where uj:RN×R→C, ψj0:RN→C for j = 1, 2, …, m and ajk = akj are positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.

Cnoidal wave solutions to Boussinesq systems

In this paper, two different techniques will be employed to study the cnoidal wave solutions of the Boussinesq systems. First, the existence of periodic travelling-wave solutions for a large family of systems is established by using a topological method. Although this result guarantees the existence of cnoidal wave solutions in a parameter region in the period and phase speed plane, it does not provide the uniqueness nor the non-existence of such solutions in other parameter regions. The explicit solutions are then found by using the Jacobi elliptic function series. Some of these explicit solutions fall in the parameter region where the cnoidal wave solutions are proved to exist, and others do not; so the method with Jacobi elliptic functions provides additional cnoidal wave solutions. In addition, the explicit solutions can be used in many ways, such as in testing numerical code and in testing the stability of these waves.

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

We study the spectral (in)stability of one-dimensional solitary and cnoidal waves of various Boussi- nesq systems. These systems model three-dimensional water waves (i.e., the surface is two-dimen- sional) with or without surface tension. We present the results of numerous computations examining the spectra related to the linear stability problem for both stationary solitary and cnoidal waves with various amplitudes, as well as multipulse solutions. The one-dimensional nature of the wave forms allows us to separate the dependence of the perturbations on the spatial variables by transverse wave number. The compilation of these results gives a full view of the two-dimensional stability problem of these one-dimensional solutions. We demonstrate that line solitary waves with elevated profiles are spectrally stable with respect to one-dimensional perturbations and long transverse perturba- tions. We show that depression solitary waves are spectrally stable with respect to one-dimensional perturbations, but unstable with respect to transverse perturbations. We also discuss the instability of multipulse solitary waves and cnoidal-wave solutions of the Boussinesq system.

The interaction of long and short waves in dispersive media

The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves seems attractive and such a system has been studied over the last decades. We evaluate the validity of this system, discussing two main problems. First, only the system coupling the linear Schrodinger equation with KdV has been derived in the literature. Second, the time variables appearing in the equations are of a different order. It appears that in the manuscripts that study the coupled NLS-KdV system, an assumption has been made that the coupled system can be derived, justifying its mathematical study. In fact, this is true even for the papers where the asymptotic derivation with the problems described above is presented. In addition to discussing these inconsistencies, we present some alternative systems describing the interaction of long and short waves.

Orbital stability of spatially synchronized solitary waves of an m-coupled nonlinear Schrödinger system

In this paper, we investigate the orbital stability of solitary-wave solutions for an m-coupled nonlinear Schrodinger system i∂∂tuj+∂2∂x2uj+∑i=1mbijui2uj=0,j=1,…,m, where m ≥ 2, uj are complex-valued functions of (x, t) ∈ ℝ2, bjj ∈ ℝ, j = 1, 2, …, m, and bij, i ≠ j are positive coupling constants satisfying bij = bji. It will be shown that spatially synchronized solitary-wave solutions of the m-coupled nonlinear Schrodinger system exist and are orbitally stable. Here, by synchronized solutions we mean solutions in which the components are proportional to one another. Our results completely settle the question on the existence and stability of synchronized solitary waves for the m-coupled system while only partial results were known in the literature for the cases of m ≥ 3 heretofore. Furthermore, the conditions imposed on the symmetric matrix B = (bij) satisfied here are both sufficient and necessary for the m-coupled nonlinear Schrodinger system to admit synchronized ground-state solutions.

Existence of periodic traveling-wave solutions for a nonlinear Schrödinger system: A topological approach

In this paper, the existence of periodic traveling-wave solutions for a nonlinear Schrodinger system is established using the topological degree theory for positive operators. The method guarantees existence of periodic solutions in a parameter region in the period and phase speed plane.