Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations

Linear stability and instability of ion-acoustic plasma solitary waves

“…The eigenspaces E u − and E s + intersect trivially since the equation is autonomous. The intersection remains trivial when adding the small perturbations in λ and ν. Exploiting exponential convergence of the coefficients of the non-autonomous terms, we may even continue the unstable subspace across the cut arg λ = π in a robust way (see [13,17,14]). This implies, that for all δ sufficiently small, we can exclude eigenvalues λ in a neighborhood of the origin.…”

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

“…The same Evans assumption has already been shown to imply long time stability of viscous profiles in the 1D case in [KK] for zero mass perturbations and [Z2,MaZ1,MaZ2,MaZ3,MaZ4,MaZ5,Z3,HZ,Ra] for general perturbations, and in the multidimensional case in [Z1, Z3, Z4] (for general perturbations); see also the important groundwork of [GZ, ZH, ZS] and [K1,K2,KS,LZe,H1,H2]. A treatment of the scalar multidimensional case (for which the Evans assumption always holds, by the maximum principle) may be found in [HoZ2,HoZ3].…”

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

“…We define the Evans function as D = W 0 , which in our setting is equivalent to the wedge-product formulations of Alexander et al (1990), Evans (1972Evans ( /1975, Zumbrun (1998), Jones (1984), and Kapitula and Sandstede (1998). More precisely, in the wedge-product formulation, we denote…”