Collapsing states of generalized Korteweg-de Vries equations

Lump Interactions and Collapse in the Modified Zakharov–Kuznetsov Equation

“…The phenomenon of wave collapse or blow-up in finite time has recently been investigated for several generalized systems. Physically, the collapse has been identified as an effective dissipation mechanism in plasma physics [14]. The question of the amplitude blow-up rate for the generalized nonlinear Schrödinger equation with cubic nonlinearity (NLS) in two and three dimensions has been investigated by numeric and asymptotic means in an important series of articles [29][30][31][32][33].…”

“…The presence of the transverse dispersion has been physically attributed to the finite Larmor radius effects [13]. The resulting two-dimensional equation in this physical context is known as the modified Zakharov-Kuznetsov (mZK) equation [14],…”

Stability and instability of periodic travelling wave solutions for the critical Korteweg–de Vries and nonlinear Schrödinger equations

“…From Theorems 2.2 and 4.1 we have that for every ε > 0, there is δ > 0 such that for all u 0 ∈ U δ , there is a T = T ( u 0 H 1 per ) > 0 and a unique solution u ∈ C ([0, T ]; H 1 per ) of (1.8) with u(x, 0) = u 0 (x) and satisfying u(t) ∈ U ε for all t ∈ [0, T ] (see [17,32,29,30]). Since…”

“…Laedke & Spatschek[29], Pelinovsky & Grimshaw[19] and the references contained therein). The main purpose of this paper is to show that there is a unique (threshold) value of the velocity c such that this value separates two different global scenarios of the evolution of a localized initial perturbation of ϕ c with respect to the periodic flow generated by (1.8).…”

Asymptotic Behavior for a Class of Solutions to the Critical Modified Zakharov-Kuznetsov Equation

“…On the other hand, the propagation of Alfvén waves at a critical angle to the undisturbed magnetic filed is described by the modified Korteweg‐de Vries (mKdV) equation For detailed description we refer to Kakutani and Ono [12]. The two dimensional generalization of this equation that describes the presence of the transverse dispersion is the mZK , see Blaha and Laedke [13]. In the physical context, this phenomenon has been attributed to the finite Larmor radius effects, Hasegawa and Uberoi [14].…”