Existence Time of Solution of the (1+2)d Knobloch Equation With Initial-Boundary Value Problem

Homoclinic and heteroclinic flows in global attractor for Davey-stewartson II equation

“…From the above lemma, (3.2) can be written as an integral equation. Then using the classical fixed theorem (see, for example, [8] or Theorem 1 in [5]), we have…”

“…The Davey-Stewartson I (DSI) and II(DSII) equations were derived by Davey-Stewartson to describe the evolution of two dimensional wave pocket on the surface of water [8] . In recent years, DSI and DSII equations have drawn much attention from many physicists and mathematicians.…”

“…(1.1) with r = 1 and r = −1 is called the DSI and DSII equations, respectively. q(x, y, t) is the amplitude of a surface wave packet, while ϕ(x, y, t) is the velocity potential of the mean flow interacting with the surface wave [8] .…”

Homoclinic degeneracy for perturbed nonlinear Schrödinger equation

“…(9) indicate that orbits will heteroclinic to the fixed periodic cycles e iθ 0 when t → ∞, namely, as t → +∞;…”

“…Obviously, u 1 (t) ∈ C ∞ (R, R) and is bounded by 1 as t → ±∞. Using the theory of [9], it is proved that u 1 (t) is uniform L ∞ bounded. Besides, |u 0 (x)| < 1 need to be proved.…”