Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group

Abstract

We first consider a nonlinear time fractional wave equation with a fractional damping posed on the Heisenberg group with Caputo fractional derivatives that interpolate the heat equation and the wave equation with the linear damping. It extends the Caputo–Wisner, the modified Szabo and the fractional Zener models. The non‐linearity accounts for a nonlinear medium. We present the Fujita exponent for blow‐up which sheds light on the admissible nonlinearity in practice. Then, we establish sufficient conditions ensuring non‐existence of local solutions. using the nonlinear capacity method. Furthermore, we extend the analysisto the case of the 2×2 system to show the flexibility of the method used.

Abundant Bounded and Unbounded Solitary, Periodic, Rogue‐Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

“…But, this method initially bifurcated the phase orbits, and different types of orbits provided different types of solutions. e solutions (33), (35), (40), and (41) are consistent with homoclinic orbits and exhibit bell wave solutions, but (34) and ( 40) yield dark bell, while (36) and ( 41) yield bright bell wave envelops.…”

“…Due to the sensitive effect of a rogue wave, scholars are being highly interested in deriving rogue and such type of colliding wave solutions in the recent exploration [33][34][35][36]. Moreover, dynamical researchers have investigated new rogon waves [37], optical M-shaped solitons with interaction with shock waves [38], orbital stability of solitary waves [39], and the nonexistence of global solutions of the time-fractional model [40] newly.…”

Nonexistence for time-fractional wave inequalities on Riemannian manifolds

“…On the other hand, in view of (40), (14) holds with f ∈ A, f (r) = exp δr 1− σ 2 for every r > 1, for some δ > 0. So by (15), F(r, θ) ≤ Cr…”

“…To the best of our knowledge, in the exception of few works on the Heisenberg group (see e.g. [15]), the issue of nonexistence for time fractional evolution inequalities on general manifolds was not previously considered in the literature. Some extensions of nonexistence results for elliptic equations and inequalities from the Euclidean case to noncompact Riemannian manifolds can be found in [11,12,13].…”

Interactions of rogue and solitary wave solutions to the (2 + 1)-D generalized Camassa–Holm–KP equation

“…Last decades, scientists are greatly astonished to various dynamical signal solutions to their riveting category of rogue-lump solutions that be capable to originate in stream bearing wave of deep ocean [13,14], plasma and nonlinear optics [15,16] as well as Bose-Einstein solidifies [17].…”