Alexei V. Penskoi
We study Lagrangian time-discretizations of the Hunter-Saxton equation. Using the Moser-Veselov approach, we obtain such discretizations defined on the Virasoro group and on the group of orientation-preserving diffeomorphisms of the circle. We conjecture that one of these discretizations is integrable.
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We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.
In this paper, we propose a two-component generalization of the generalized Hunter-Saxton equation obtained in \cite{BLG2008}. We will show that this equation is a bihamiltonian Euler equation, and also can be viewed as a bi-variational equation.
On a new 3D generalized Hunter-Saxton equation
研究三维广义Hunter-Saxton方程的可积性，通过变换化为常特征形式并平凡化Lax表示，最终证明其不通过Painleve检验。
We consider N=2 supersymmetric extensions of the Camassa-Holm and Hunter-Saxton equations. We show that they admit geometric interpretations as Euler equations on the superconformal algebra of contact vector fields on the 1|2-dimensional supercircle. We use the bi-Hamiltonian formulation to derive Lax pairs. Moreover, we present some simple examples of explicit solutions. As a by-product of our analysis we obtain a description of the bounded traveling-wave solutions for the two-component Hunter-Saxton equation.
In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its $μ$-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group $\Diff(§)$ with a space of scalar functions on $§$ we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2$μ$HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [J. Esc…
The generalized Hunter-Saxton system comprises several well-known models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in $L^p$, $p \in [1,\infty)$, spaces for nonzero real parameters $(λ,κ)$. Our results significantly improve/extend those by Wunsch et al. [27-29] and Sarria [21]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a…
We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the …
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the $μ$--Hunter--Saxton equation, used to model liquid crystals under influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray. The descending property of the metrics is used to obtain a novel factorisation of diffeomo…
Exact Analytic Solution for the Time-Fractional Hunter-Saxton Equation with Caputo derivative
该研究将Hunter-Saxton方程推广至分数阶时间导数，利用分离变量法得到首个精确闭式解，为数值方法提供基准并揭示分数阶记忆对非线性波传播的影响。
Long-time asymptotic behavior of the Hunter-Saxton equation
本研究利用推广的Deift-Zhou最速下降法，通过Riemann-Hilbert问题变形，得到Hunter-Saxton方程柯西问题解在两个时空区域的长时间渐近近似，明确了不同区域的衰减速率与误差阶。
In this paper, we present infinitely many conserved densities satisfying particular conservation law $F_{t}=(2uF)_{x}$ for the generalized Riemann equations at $N=2,3,4$. In the $N=2$ case, we also construct conserved densities corresponding to new conservation laws containing an arbitrary smooth function. In virtue of reductions and/or changes of variables, related conserved densities are obtained for two component Hunter-Saxton equation, Hunter-Saxton equation, Gurevich-Zybin equation and Monge-Ampere equation.
A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.
The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.
We show that the continuous limit of a wide natural class of the right-invariant discrete Lagrangian systems on the Virasoro group gives the family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and Korteweg-de Vries equations. This family has been recently derived by Khesin and Misiolek as Euler equations on the Virasoro algebra for $H^1_{α,β}$-metrics. Our result demonstrates a universal nature of these equations.
In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equation and the Euler-Poisson equation.
Symmetry groups of geodesic equations with applications in water waves
本文通过将水波和液晶中的关键方程表示为无限维群上右不变度量的测地线方程，研究其对称群，并利用度量和拓扑修正改善Hopf方程行为，为数值方法提供基准。
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with $2$-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.
We show that the following three systems related to various hydrodynamical approximations: the Korteweg--de Vries equation, the Camassa--Holm equation, and the Hunter--Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sid…
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