scholarly journals Nonexistence results for a fully nonlinear evolution inequality

2016 ◽  
Vol 23 (0) ◽  
pp. 19-24
Author(s):  
Qianzhong Ou
Keyword(s):  
Nonlinear Evolution ◽  
Fully Nonlinear ◽  
Evolution Inequality

scholarly journals On a discrete scheme for time fractional fully nonlinear evolution equations

2020 ◽  
Vol 120 (1-2) ◽  
pp. 151-162 ◽  
Author(s):  
Yoshikazu Giga ◽  
Qing Liu ◽  
Hiroyoshi Mitake
Keyword(s):  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Discrete Scheme ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Convergence Of The Scheme

We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo’s time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.

scholarly journals Nonlinear evolution of the modulational instability under weak forcing and damping

2010 ◽  
Vol 10 (12) ◽  
pp. 2589-2597 ◽  
Author(s):  
J. Touboul ◽  
C. Kharif
Keyword(s):  
Viscous Dissipation ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wind Forcing ◽  
Marginal Stability ◽  
Breaking Wave ◽  
Wind Model ◽  
Fully Nonlinear ◽  
Nonlinear Potential ◽  
Frequency Downshifting

Abstract. The evolution of modulational instability, or Benjamin-Feir instability is investigated within the framework of the two-dimensional fully nonlinear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles' theory. The introduction of dissipation in the equations is briefly discussed. Evolution of this instability in the presence of damping was considered by Segur et al. (2005a) and Wu et al. (2006). Their results were extended theoretically by Kharif et al. (2010) who considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon.

scholarly journals Evolution and Stability of Two-Dimensional Anelastic Internal Gravity Wave Packets

2018 ◽  
Vol 75 (10) ◽  
pp. 3703-3724 ◽  
Author(s):  
Alain D. Gervais ◽  
Gordon E. Swaters ◽  
Ton S. van den Bremer ◽  
Bruce R. Sutherland
Keyword(s):  
Wave Packet ◽  
Gravity Wave ◽  
Linear Theory ◽  
Nonlinear Evolution ◽  
Wave Packets ◽  
Internal Gravity Wave ◽  
Two Dimensional ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Nonlinear Simulations

The weakly nonlinear evolution, stability, and overturning of horizontally and vertically localized internal gravity wave packets is examined for a nonrotating, anelastic atmosphere that is stationary in the absence of waves. The weakly nonlinear evolution is examined through the derivation of their wave-induced mean flow, which is used to formulate a nonlinear Schrödinger equation. The induced flow is manifest as a long, hydrostatic, bow wake-like disturbance, whose flow direction transitions from positive on the leading flank of the wave packet to negative on the trailing flank of the wave packet. As such, two-dimensional wave packets are always modulationally unstable. This instability results in enhanced amplitude growth confined to either the leading or trailing flank. Hence, when combined with anelastic growth predicted by linear theory, we anticipate two-dimensional waves will overturn either somewhat below or just above the heights predicted by linear theory. Numerical solutions of the Schrödinger equation are compared with the results of fully nonlinear simulations to establish the validity of the weakly nonlinear theory. Actual wave overturning heights are determined quantitatively from a range of fully nonlinear simulations.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

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