Nonautonomous matter-wave solitons near the Feshbach resonance

Nonperiodic Oscillation of Bright Solitons in Condensates with a Periodically Oscillating Harmonic Potential 10.5560/ZNA.2012-0085

“…Meanwhile, the bright soliton propagates along the positive direction of the x-axis. This phenomenon is similar to that for k = 0 in [14]. This suggests that the propagation properties of bright solitons are hardly dependent on the fast oscillating potential.…”

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

“…In the second example, we select γ 1 (t) = 1, γ 2 (t) = 1, γ 3 (t) = β, a 0 = −2 α + β 2 , k 1 = 4, l 1 = −2 and set the integration constants of (3.16) and (3.17) as −θ 13 /6 √ 2, then the one-soliton solutions (3.18) and (3.19) give the known solutions [27] where A and B satisfy the constant-coefficient AKNS equations In view of (2.11) and (3.1), we reduce (3.2) and (3.3) as In the procedure of extending Hirota's bilinear method to (1.1) and (1.2), one of the key steps is to reduce (1.1) and (1.2) to the bilinear forms (2.9) and (2.10) by the transformations (2.7), (2.8) and (2.11). It is graphically shown that the dynamical evolutions of one-soliton solutions (3.18) and (3.19), two-soliton solutions (3.20) and (3.21), three-soliton solutions (3.22) and (3.23) possess time-varying amplitudes as Serkin et al [37][38][39] reported in the process of propagations. Recently, fractional-order differential calculus and its applications have attached much attention [3,26,54,71].…”

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics

“…It should be stressed that to test the validity of our predictions, the experimental arrangement should be inspected to be as close as possible to the optimal map of parameters, at which the problem proves to be exactly integrable (Serkin & Hasegawa, 2000a;2002). Notice, that when Serkin and Hasegawa formulated their concept of solitons in nonautonomous systems (Serkin & Hasegawa, 2000a;2002), known today as nonautonomous solitons and SH-theorems (Serkin & Hasegawa, 2000a;2002) published for the first time in (Serkin & Hasegawa, 2000a;2002), they emphasized that "the methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark "soliton islands" in a "sea of solitary waves" with varying dispersion, nonlinearity, and gain or absorption" (Belyaeva et al, 2011;Serkin et al, 2010a;. The concept of nonautonomous solitons, the generalized Lax pair and generalized AKNS methods described in details in this Chapter can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter-waves and offer many opportunities for further scientific studies.…”