scholarly journals Local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form

2017 ◽  
Vol 446 (2) ◽  
pp. 1606-1631 ◽  
Author(s):  
D. Catalano Ferraioli ◽  
L.A. de Oliveira Silva
Keyword(s):  
Conservation Law ◽  
Evolution Equations ◽  
Isometric Immersions ◽  
Pseudospherical Surfaces

scholarly journals Third-order differential equations and local isometric immersions of pseudospherical surfaces

2016 ◽  
Vol 18 (06) ◽  
pp. 1650021 ◽  
Author(s):  
Tarcísio Castro Silva ◽  
Niky Kamran
Keyword(s):  
Differential Equations ◽  
Fundamental Form ◽  
Evolution Equations ◽  
Gordon Equation ◽  
Second Fundamental Form ◽  
Isometric Immersions ◽  
Sine Gordon Equation ◽  
Third Order ◽  
The Second Fundamental Form ◽  
Pseudospherical Surfaces

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and nonlocal) and the presence of associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa–Holm and Degasperis–Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in E3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class introduced by Chern and Tenenblat [Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986) 55–83]. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the partial differential equation. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In a pair of earlier papers [N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, to appear in Comm. Anal. Geom., arXiv: 1308.6545; Local isometric immersions of pseudo-spherical surfaces and evolution equations, in Hamiltonian Partial Differential Equations and Applications, eds. P. Guyenne, D. Nichols and C. Sulem, Fields Institute Communications, Vol. 75 (Springer-Verlag, 2015), pp. 369–381], it was shown that this property fails to hold for all [Formula: see text]th-order evolution equations [Formula: see text] and all other second-order equations of the form [Formula: see text], except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of [Formula: see text] and [Formula: see text] which are independent of the choice of solution [Formula: see text]. In this paper, we consider third-order equations of the form [Formula: see text], [Formula: see text], which describe pseudospherical surfaces with the Riemannian metric given in [T. Castro Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differential Equations 259 (2015) 4897–4923]. This class contains the Camassa–Holm and Degasperis–Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E3 for which the coefficients of the second fundamental form depend on a jet of finite order of [Formula: see text], then these coefficients are universal in the sense of being independent on the choice of solution [Formula: see text]. This result further underscores the special place that the sine-Gordon equations seem to occupy amongst integrable partial differential equations in one space variable.

scholarly journals Local isometric immersions of pseudo-spherical surfaces and kth order evolution equations

2019 ◽  
Vol 21 (04) ◽  
pp. 1850025
Author(s):  
Nabil Kahouadji ◽  
Niky Kamran ◽  
Keti Tenenblat
Keyword(s):  
Isometric Immersion ◽  
Evolution Equations ◽  
Second Fundamental Form ◽  
Second Order ◽  
Isometric Immersions ◽  
Spherical Surfaces ◽  
Pseudospherical Surfaces ◽  
The Mean ◽  
Local Isometric Immersion ◽  
Straight Lines

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.

scholarly journals Fourth order evolution equations which describe pseudospherical surfaces

2014 ◽  
Vol 257 (9) ◽  
pp. 3165-3199 ◽  
Author(s):  
Diego Catalano Ferraioli ◽  
Keti Tenenblat
Keyword(s):  
Fourth Order ◽  
Evolution Equations ◽  
Pseudospherical Surfaces

scholarly journals Diffusion Processes Satisfying a Conservation Law Constraint

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
J. Bakosi ◽  
J. R. Ristorcelli
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Conservation Law ◽  
Evolution Equations ◽  
Diffusion Processes ◽  
Random Variables ◽  
Planck Equation ◽  
Equation Model ◽  
First Four Moments ◽  
And Diffusion

We investigate coupled stochastic differential equations governing N nonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

scholarly journals Second order evolution equations which describe pseudospherical surfaces

2016 ◽  
Vol 260 (11) ◽  
pp. 8072-8108 ◽  
Author(s):  
D. Catalano Ferraioli ◽  
L.A. de Oliveira Silva
Keyword(s):  
Evolution Equations ◽  
Second Order ◽  
Pseudospherical Surfaces

Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

2014 ◽  
Vol 760 ◽  
pp. 368-386 ◽  
Author(s):  
Alexei F. Cheviakov ◽  
Martin Oberlack
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Infinite Family ◽  
Time Dependent ◽  
Local Conservation ◽  
Direct Construction ◽  
Primitive Variables

AbstractLocal conservation laws are systematically constructed for three-dimensional time-dependent viscous and inviscid incompressible fluid flows, in primitive variables and vorticity formulation, using the direct construction method. Complete sets of local conservation laws in primitive variables are derived for the case of conservation law multipliers depending on derivatives up to the second order. In the vorticity formulation, there exists an infinite family of vorticity-dependent conservation laws involving an arbitrary differentiable function of space and time, holding for both viscous and inviscid cases. The infinite conservation law family is used to generate further independent hierarchies of conservation laws that essentially involve vorticity and arbitrary flow parameters, which are determined by known evolution equations such as those for momentum, energy or helicity, though not necessarily in the form of a conservation law. The new conservation laws are not restricted to any reduced flow geometry such as planar or axisymmetric limits. Examples are considered.

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