scholarly journals On nonlocal symmetries generated by recursion operators: Second-order evolution equations

2017 ◽  
Vol 37 (8) ◽  
pp. 4239-4247 ◽  
Author(s):  
M. Euler ◽  
◽  
N. Euler ◽  
M. C. Nucci ◽  
Keyword(s):  
Evolution Equations ◽  
Second Order ◽  
Recursion Operators ◽  
Nonlocal Symmetries

Nonformal recursion operators and nonlinear integrable evolution equations

1986 ◽  
Vol 12 (2) ◽  
pp. 97-100 ◽  
Author(s):  
M. Boiti ◽  
B. G. Konopelchenko
Keyword(s):  
Evolution Equations ◽  
Recursion Operators ◽  
Integrable Evolution

A Method to Construct the Nonlocal Symmetries of Nonlinear Evolution Equations

2013 ◽  
Vol 30 (10) ◽  
pp. 100202 ◽  
Author(s):  
Xiang-Peng Xin ◽  
Yong Chen
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Nonlocal Symmetries

Subharmonic resonance of short internal standing waves by progressive surface waves

1996 ◽  
Vol 321 ◽  
pp. 217-233 ◽  
Author(s):  
D. F. Hill ◽  
M. A. Foda
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Growth Rates ◽  
Evolution Equations ◽  
Standing Waves ◽  
Second Order ◽  
Inviscid Limit ◽  
Initial Growth

Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.

Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor

2014 ◽  
Vol 1651 ◽  
Author(s):  
Thomas Hochrainer
Keyword(s):  
Dislocation Density ◽  
Evolution Equations ◽  
Continuum Theory ◽  
Orientation Distribution ◽  
Line Length ◽  
Second Order ◽  
Dislocation Dynamics ◽  
Alignment Tensor ◽  
Length Changes ◽  
Edge Dislocations

ABSTRACTIn the current paper we present a continuum theory of dislocations based on the second-order alignment tensor in conjunction with the classical dislocation density tensor (Kröner-Nye-tensor) and a scalar dislocation curvature measure. The second-order alignment tensor is a symmetric second order tensor characterizing the orientation distribution of dislocations in elliptic form. It is closely connected to total densities of screw and edge dislocations introduced in the literature. The scalar dislocation curvature density is a conserved quantity the integral of which represents the total number of dislocations in the system. The presented evolution equations of these dislocation density measures partly parallel earlier developed theories based on screw-edge decompositions but handle line length changes and segment reorientation consistently. We demonstrate that the presented equations allow predicting the evolution of a single dislocation loop in a non-trivial velocity field.

Remarks on Controllability of Second Order Evolution Equations in Hilbert Spaces

1970 ◽  
Vol 8 (1) ◽  
pp. 90-99 ◽  
Author(s):  
Kunio Tsujioka
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Second Order

scholarly journals IDENTIFICATION PROBLEMS FOR THE SYSTEM GOVERNED BY ABSTRACT NONLINEAR DAMPED SECOND ORDER EVOLUTION EQUATIONS

2004 ◽  
Vol 41 (3) ◽  
pp. 435-459 ◽  
Author(s):  
Jun-Hong Ha ◽  
Shin-Ichi Nakagiri
Keyword(s):  
Evolution Equations ◽  
Second Order ◽  
Identification Problems
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