The modified Hunter–Saxton equation

Górka, Przemysław; Reyes, Enríque G.

doi:10.1016/j.geomphys.2012.03.011

Cited by 16 publications

(20 citation statements)

References 22 publications

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“…When k = 2, the proof of this proposition was given in [14]. Therefore, we conclude that when f 31 = λf 11 = 0, λ 2 = 1 for any k ≥ 2, the system of equations (11), (12) and (13) is inconsistent.…”

Section: Proof Of Theoremmentioning

confidence: 77%

“…Differentiating the latter two equations with respect to z k−1 leads to (a − c)ηf 32,z k−1 = 0, 2bηf 32,z k−1 = 0, and since f 32,z k−1 = 0, we conclude that a−c = b = 0, which contradicts the Gauss equation (13). Therefore, for any k ≥ 2, when f 11 = 0, the system of equations (11), (12) and (13) is inconsistent.…”

Section: Proof Of Theoremmentioning

confidence: 81%

“…Proposition 3. For k-th evolution equations, k ≥ 2, describing η pseudo-spherical surfaces with associated 1-forms ω i = f i1 dx + f i2 dt, 1 ≤ i ≤ 3, where either f 11 = 0 or f 31 = 0, the system of equations (11), (12) and (13) is inconsistent.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

Kahouadji

Kamran

Tenenblat

2015

Hamiltonian Partial Differential Equations and Applications

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We consider the class of evolution equations that describe pseudo-spherical surfaces of the form ut = F (u, ∂u/∂x, ..., ∂ k u/∂x k ), k ≥ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature -1. We investigate the following problem: given such a metric, is there a local isometric immersion in R 3 such that the coefficients of the second fundamental form of the surface depend on a jet of finite order of u? By extending our previous result for second order evolution equation to k-th order equations, we prove that there is only one type of equations that admit such an isometric immersion. We prove that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions u are universal, i.e., they are independent of u. 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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Section: Proof Of Theoremmentioning

confidence: 77%

Section: Proof Of Theoremmentioning

confidence: 81%

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Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

Kahouadji

Kamran

Tenenblat

2015

Hamiltonian Partial Differential Equations and Applications

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“…It is also shown in [5] that one can generate infinite sequences of conservation laws for the class of differential equations describing η pseudospherical surfaces by making use of the structure equations (3), although some of these conservation laws may end up being non-local. Important further developments of these ideas around this theme can be found in [15], [16], [17], [18], [12], [19], [20], [7], [9], [10], [11], [8], [21]. One may also consider the class of differential equations describing pseudospherical surfaces from an extrinsic point of view, motivated by the classical result which says that every pseudospherical surface can be locally isometrically immersed in E 3 .…”

Section: Introductionmentioning

confidence: 99%

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

Silva

Kamran

2016

Commun. Contemp. Math.

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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 non-local) and the presence 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 E 3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class of Chern and Tenenblat [3]. 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 pde. 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 [13], [14] we have shown that this property fails to hold for all k-th order evolution equations ut = F (u, ux, ..., u x k ) and all other second order equations of the form uxt = F (u, ux), 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 x and t which are independent of the choice of solution u. In the present paper, we consider thirdorder equations of the form ut − uxxt = λuuxxx + G(u, ux, uxx), λ ∈ R, which describe pseudospherical surfaces. This class contains the Camassa-Holm and Degasperis-Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E 3 for which the coefficients of the second fundamental form depend on a jet of finite order of u, then these coefficients are universal in the sense of being independent on the choice of solution u. This result further underscores the special place that the sine-Gordon equations seems to occupy amongst integrable partial differential equations in one space variable.

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“…Following the method in Ref. [10,11,16,19,27], one can get an infinite number of nonlocal symmetries for the MGLDW system (2.3)-(2.4). To this end, we consider (2.6) and (2.8) in Lax pairs as 14) where the eigenfunctions ψ and φ in Lax pairs are λ dependent, and the fields m and n are λ independent.…”

Section: Nonlocal Symmetrymentioning

confidence: 99%

Nonlocal symmetry constraints and exact interaction solutions of the (2+1) dimensional modified generalized long dispersive wave equation

Chen¹,

Chen²

2021

JNMP

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In this paper, nonlocal symmetry of the (2+1) dimensional modified generalized long dispersive wave system and its applications are investigated. The nonlocal symmetry related to the eigenfunctions in Lax pairs is derived, and infinitely many nonlocal symmetries are obtained. By introducing three potentials, the prolongation is found to localize the given nonlocal symmetry. Various finite-and infinite-dimensional integrable models are constructed by using the nonlocal symmetry constraint method. Moreover, applying the general Lie symmetry approach to the enlarged system, the finite symmetry transformation and similarity reductions are computed to give novel exact interaction solutions. In particular, the explicit soliton-cnoidal wave solution is obtained for the modified generalized long dispersive wave system, and it can be reduced to the two-dark-soliton solution in one special case.

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The modified Hunter–Saxton equation

Cited by 16 publications

References 22 publications

Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

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

Nonlocal symmetry constraints and exact interaction solutions of the (2+1) dimensional modified generalized long dispersive wave equation

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