Second order evolution equations which describe pseudospherical surfaces

We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

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Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2010024 ◽

2010 ◽

Vol 17 (3) ◽

pp. 836-857 ◽

Cited By ~ 16

Author(s):

Hedy Attouch ◽

Paul-Émile Maingé

Keyword(s):

Asymptotic Behavior ◽

Evolution Equations ◽

Second Order

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Subharmonic resonance of short internal standing waves by progressive surface waves

Journal of Fluid Mechanics ◽

10.1017/s0022112096007707 ◽

1996 ◽

Vol 321 ◽

pp. 217-233 ◽

Cited By ~ 24

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.

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Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor

MRS Proceedings ◽

10.1557/opl.2014.53 ◽

2014 ◽

Vol 1651 ◽

Cited By ~ 3

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.

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