On the normalized ground states for the Kawahara equation and a fourth order NLS

Abstract

We consider the Kawahara model and two fourth order semi-linear Schrödinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable. For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. In fact, our results establish (modulo an additional technical assumption, which should be satisfied at least generically), spectral stability for all normalized waves constructed thereinin all dimensions, for all acceptable values of the parameters. This, combined with the results of [5], provides orbital stability, for all normalized waves enjoying the non-degeneracy property. The validity of the non-degeneracy property for generic waves remains an intriguing open question. At the same time, we verify and clarify recent numerical simulations of the spectral stability of these solitons. For the fourth order NLS models, we improve upon recent results on spectral stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order anisotropic NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.

Nanoptera In Higher-Order Nonlinear Schrödinger Equations: Effects Of Discretization

“…where gpλq and Λpλq are the only parts of this expression that depend on the discretization. The prefactor of the continuous fourth-order Karpman equation ( 2) also has the form in (60). By comparing the expression in ( 60) with ( 24), we see that gpλq " ´V in the prefactors of the series terms in the continuous fourth-order Karpman equation.…”

“…One can show that Λpλq in (60) tends to the corresponding value for the continuous equation ( 2) by following the same steps as the inner analysis in Appendix B.1. One can also show that increasing the order 2q of the approximation removes terms from the inner analysis.…”

“…that fourth-order dispersion can produce both bright [1,5,16,31,32,35,36,38,58,60,64] and dark [2] solitary waves in photonic systems, whereas third-order dispersion prevents stable solitary-wave behavior [78]. Solitary waves have also been observed in experiments in systems that are modeled by sixth-order, eighth-order, and tenth-order dispersion, and they have been studied numerically for even dispersion orders up to sixteenth-order dispersion [63].…”

Kink-Antikink Interaction Forces and Bound States in a $ϕ^4$ Model with Quadratic and Quartic dispersion

“…Not only has this type of dispersion engineering been realized in the lab, but it has also been used in the context of the so-called pure-quartic soliton laser [28]. Mathematical studies of the existence and stability of solitary waves of such equations are also ongoing [29][30][31].…”

Dark solitons under higher order dispersion

“…All of these bright solitons exist when the highest-order of dispersion is negative. Concurrently, this has inspired substantial mathematical interest in the analysis of such models [6][7][8][9].…”