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

On the nonlinear self-adjointness of a class of fourth-order evolution equations

2016 ◽  
Vol 275 ◽  
pp. 299-304 ◽  
Author(s):  
R. Tracinà ◽  
M.S. Bruzón ◽  
M.L. Gandarias
Keyword(s):  
Fourth Order ◽  
Evolution Equations

Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves

2002 ◽  
Vol 450 ◽  
pp. 201-205 ◽  
Author(s):  
ELIEZER KIT ◽  
LEV SHEMER
Keyword(s):  
Gravity Waves ◽  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Velocity Potential ◽  
Surface Elevation ◽  
Two Dimensional ◽  
Zakharov Equation ◽  
Dimensional Version

A spatial two-dimensional version of the Zakharov equation describing the evolution of deep-water gravity waves is used to derive two fourth-order evolution equations, for the amplitudes of the surface elevation and of the velocity potential. The scaled form of the equations is presented.

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

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 Fourth order quasilinear evolution equations of hyperbolic type

1992 ◽  
Vol 44 (4) ◽  
pp. 619-630 ◽  
Author(s):  
Alberto AROSIO ◽  
Roberto NATALINI ◽  
Maria Gabriella PAOLI
Keyword(s):  
Fourth Order ◽  
Evolution Equations ◽  
Hyperbolic Type
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