We consider integrability structures of the generalized Hunter--Saxton equation. In particular, we obtain the Lax representation with nonremovable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give an example of employing the higher order symmetry to constructing exact globally defined solutions for the generalized Hunter--Saxton equation.
It is shown that two different supersymmetric extensions of the Harry Dym equation lead to two different negative hierarchies of the supersymmetric integrable equations. While the first one yields the known even supersymmetric Hunter - Saxton equation, the second one is a new odd supersymmetric Hunter - Saxton equation. It is further proved that these two supersymmetric extensions of the Hunter - Saxton equation are reciprocally transformed to two different supersymmetric extensions of the Liouville equation.
Exact rational solutions of the generalized Hunter-Saxton equation are obtained using Padé approximant approach for the traveling-wave and self-similarity reduction. A larger class of algebraic solutions are also obtained by extending a range of parameters within the solutions obtained by this approach.
In this paper, we study the bilinear form and the general N-soliton solution for a two-component Hunter-Saxton (2-HS) equation, which is the short wave limit of a twocomponent Camassa-Holm equation. By defining a hodograph transformation based on a conservation law and appropriate dependent variable transformations, we propose a set of bilinear equations which yields the 2-HS equation. Furthermore, we construct the N-soliton solution to the 2-HS equation based on the tau functions of an extended two-dimensional Toda-lattice hierarchy through reductions. One- and two-soliton solutions are calcu…
In this paper, we present extraordinary algebraic and geometrical structures for the Hunter-Saxton equation: infinitely many commuting and non-commuting $x,t$-independent higher order symmetries and conserved densities. Using a recursive relation, we explicitly generate infinitely many higher order conserved densities dependent on arbitrary parameters. We find three Nijenhuis recursion operators resulting from Hamiltonian pairs, of which two are new. They generate three hierarchies of commuting local symmetries. Finally, we give a local recursion operator depending on an arbitrary parameter. A…
This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the Hunter-Saxton (HS) hierarchy through studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the HS hierarchy, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the Baker-Ahhiezer functions, th…
This paper is dedicated to provide the global solutions of algebro-geometric type for all the equations of a new commuting hierarchy containing the Hunter-Saxton (HS) equation. Our main tools include the zero curvature method to derive the HS hierarchy, the generalized Jacobian variety, the generalized Riemann theta function, the Weyl $m$-fucntions $m_\pm(x,t,z)$, and the pole motion obtained by solving an inverse problem for the Sturm-Liouville equation $L(ψ_1)=-ψ_1^{\prime\prime}=zyψ_1$. Based on these tools and the theory of nonautonomous differential systems, topological dynamics and ergod…
In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation.
This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtain…
Through a reciprocal transformation $\mathcal{T}_0$ induced by the conservation law $\partial_t(u_x^2) = \partial_x(2uu_x^2)$, the Hunter-Saxton (HS) equation $u_{xt} = 2uu_{2x} + u_x^2$ is shown to possess conserved densities involving arbitrary smooth functions, which have their roots in infinitesimal symmetries of $w_t = w^2$, the counterpart of the HS equation under $\mathcal{T}_0$. Hierarchies of commuting symmetries of the HS equation are studied under appropriate changes of variables initiated by $\mathcal{T}_0$, and two of these are linearized while the other is identical to the hierar…
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.
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.
The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the $H^1$ and $\dot{H}^1$ right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The CH and HS equations are integ…
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
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.
Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases $%N=2,3$ and N=4 for which the conservation laws, Lax type representations and bi-Hamiltonian structures are analyzed in detail. We also show that the case N=2 is equivalent to a generalized Hunter-Saxton dynamical system, whose integrability …
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.
In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.
Zero curvature formulations, pseudo-potentials, modified versions, ``Miura transformations'', and nonlocal symmetries of the Korteweg--de Vries, Camassa-- Holm and Hunter--Saxton equations are investigated from an unified point of view: these three equations belong to a two--parameters family of equations ``describing pseudo-spherical surfaces'', and therefore their basic integrability properties can be studied by geometrical means.
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.
