Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities

Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schrödinger equation

“…This theorem implies in particular that stability change occurs at most once, which is conjectured in [31, p. 265], and is in contrast to NLS with triple power nonlinearity considered in [30].…”

“…when q = 2p − 1, see e.g. [30]), there does not exist an explicit formula for the full standing waves profile. Note that ω * = ∞ when a q > 0, 0 < ω * < ∞ when a p > 0 and a q < 0, and ω * = −∞ (i.e.…”

Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schr{ö}dinger equation

“…In particular, in the cases 1(c), 2(b) and 3(b) with stability change, the standing wave φ ωc at the critical frequency ω c is unstable. This theorem implies in particular that stability change occurs at most once, which is conjectured in [30, p. 265], and is in contrast to NLS with triple power nonlinearity considered in [29].…”

“…when q = 2p − 1, see e.g. [29]), there does not exist an explicit formula for the full standing waves profile. Note that ω * = ∞ when a q > 0, 0 < ω * < ∞ when a p > 0 and a q < 0, and ω * = −∞ (i.e.…”

“…We have performed simulations for (p, q) on the line q = 2p − 1, as for this range of exponents explicit formulas are available for solitary wave profiles (see e.g. [29]) and can be used easily to construct initial data. Considering other ranges of (p, q) would have been possible, to the extend of additional computations to first obtain numerically solitary waves.…”