On a nonstationary nonlinear coupled system

Abstract

In this paper, a strongly nonlinear coupled elliptic-parabolic system modelling a class of engineering problems with heat effect is studied. Existence of a weak solution is first established by Schauder fixed point theorem, where the coupled functions &#963;(s), k(s) are assumed to be bounded. The uniqueness of the solution is obtained by applying Meyers' theorem and assuming that &#963;(s), k(s) are Lipschitz continuous. The regularity of the solution is then analyzed in dimension d < 2 under the assumptions on &#963;(s), k(s) &#8712; C2(R) and the boundedness of their derivatives of second order. Finally, the blow-up phenomena of the system are studied. Mathematical subject classification: 35J60, 35K55

Real‐Time Observation of Double‐Hopf Bifurcation in an Ultrafast All‐PM Fiber Laser

“…To our knowledge, this is the first experimental observation of such phenomenon in lasers and one of a few experimental reports in other physical systems, such as in liquid flows [ 17 ] or some simple electronic and mechanical setups. [ 18 ] Indeed, although the presence of double‐Hopf bifurcation has been extensively studied theoretically [ 19–21 ] and even predicted for class‐B laser subjected to delayed feedback, [ 22 ] its measurement examples can only rarely be found. [ 17,18 ]…”

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov–Kuznetsov Equation

“…Because many important phenomena in various fields such as physics, mechanics, biology, chemistry, and engineers can be described by NLPDEs. In order to obtain the exact solutions of NLPDEs, many powerful methods have been developed such as Hirotas bilinear method, [1] extended tanh-function method, [2] The sine-cosin method, [3] Exp-function method, [4] Jacobi elliptic function expansion method, [5] the modified extended tanh-function method, [6] (G ′ /G)-expansion method, [7] etc. In this paper, the work is to establish wave solutions of the generalized (2+1)-dimensional nonlinear Z-K equation given by…”