sine-Gordon Hierarchy Research Articles

Algebro-Geometric Solutions of the Sine-Gordon Hierarchy

On the basis of two sets of Lenard recursion sequences and zero-curvature equation associated with a matrix spectral problem, we derive the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.

Hamiltonian multiform description of an integrable hierarchy

Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by the results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable 1 + 1-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as the starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalization of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg–de Vries hierarchy, the sine-Gordon hierarchy (in light-cone coordinates), and the Ablowitz–Kaup–Newell–Segur hierarchy.

Conservation laws and Hamiltonian structures of the generalized sine-Gordon hierarchy

By introducing a 2 × 2 matrix spectral problem, a new hierarchy of nonlinear evolution equations is proposed. A typical equation in this hierarchy is the generalization of sine-Gordon equation. With the aid of trace identity, the Hamiltonian structures of the hierarchy are constructed. In addition, the infinite sequence of conserved quantities of the generalized sine-Gordon equation are obtained.

The exact solutions for the nonisospectral AKNS hierarchy through the inverse scattering transform

Exact solutions for the nonisospectral AKNS hierarchy are obtained through the inverse scattering transform. Some explicit forms of these solutions are further given. Reductions are also considered. Exact solutions of some nonisospectral hierarchies, such as the nonisospectral mKdV hierarchy, the nonisospectral KdV hierarchy, the nonisospectral nonlinear Schrödinger hierarchy and the nonisospectral sine-Gordon hierarchy, are obtained.

Vanishing residue characterization of the sine-Gordon hierarchy

Vanishing residue characterization of the sine-Gordon hierarchy

Vanishing residue characterization of the sine-Gordon hierarchy

The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.

On the characterization of the nonlinear Schrödinger hierarchy

It is proved that the conserved polynomials of the nonlinear Schrodinger equation have a vanishing residue property analogous to those now known to characterize the Korteweg-de Vries, Modified Korteweg-de Vries and Sine-Gordon hierarchies.

Fused Hierarchy Produced by mKdV Hierarchy and Sine-Gordon Hierarchy

By using the inverse problem, we have found a system with two hierarchies, that is, the localized induction hierarchy and the hierarchy including the dispersionless equation. Recently, with the same system, we have found a fused hierarchy which is produced by fusing two hierarchies with help of the recursion formulae. In this short note, we would like to show that the mKdV hierarchy and the sine-Gordon hierarchy produce another fused hierarchy. It is recognized that the recursion operators of the mKdV hierarchy and the sine-Gordon hierarchy is the inverse operator in each other. However, it has been misunderstood that two hierarchies are not connected since the boundary conditions have not been carefully considered. If each member of equations in the mKdV hierarchy is written by tðx; tÞ nðx; tÞ 1⁄4 0; ðn 1⁄4 1; 2; . . .Þ ð1Þ

Recursion formulae for IKK system with two hierarchies

In a previous paper an inverse system with two hierarchies for a given symmetry was proposed. The scheme was found to have so constructive structure that relation between members of the hierarchies was systematically determined. In this Letter, with the proposed system we find simply and directly recursion formulae for two hierarchies for the su(2) algebra, i.e., the localized induction hierarchy and the hierarchy with the dispersionless equation. We show that, as a special case, the recursion formulae reduce to those of the modified Korteweg–de Vries hierarchy and the sine–Gordon hierarchy.

Integrable models constructed from the symmetries of the modified KdV equation

Starting from the symmetries of the modified KdV (MKdV) equation, Liouville, sinh-Gordon and/or sine-Gordon hierarchies can be established besides the MKdV hierarchy and the KdV hierarchy. The Liouville, sinh-Gordon and/or sine-Gordon hierarchies can be considered as one half hierarchy of the generalized MKdV hierarchy can be considered as another half hierarchy of the generalized MKdV hierarchy.

Gauge and B�cklund transformations for the generalized sine-Gordon equation and its ?-dependent modified equation

We study the generalized sine-Gordon hierarchy and its associatedη-dependent modified sine-Gordon hierarchy. Two Backlund transformations for these two families are constructed. One of them is a generalization of the Backlund transformations of Wadatiet al. and the other one is new. Gauge transformations of a relevant AKNS system are employed to reduce the integration of these equations via the Backlund transformations to quadratures. Three generations of explicit solutions of the sine-Gordon equation are presented.

Geometric theory of local and non-local conservation laws for the sine-Gordon equation

In previous work one of the authors gave a geometric theory of those nonlinear evolution equations (n. e. es) that can be solved by the Zakharov & Shabat (1972) inverse scattering scheme as generalized by Ablowitz, Kaup, Newell & Segur (1973 b , 1974). In this paper we extend the geo­metric theory to include the Hamiltonian structure of those n. e. es solvable by the method, and we indicate the connection between the geometric theory and the theory of prolongation structures and pseudopotentials due to Wahlquist & Estabrook (1975, 1976). We exploit a ‘gauge’ in­variance of the geometric theory to derive both the well known polynomial conserved densities of the sine-Gordon equation and a non-local set of conserved densities. These act as Hamiltonian densities for a hierarchy of sine-Gordon equations which is analogous to that found by Lax (1968) for the Korteweg-de Vries equation and appears to be new. In an Appendix we derive an expression for the equation of motion for an arbitrary member of the sine-Gordon hierarchy by methods which can be applied in larger context. The results for the sine-Gordon equation lead to the conclusion that a complete set of conserved densities for an arbitrary n. e. e. solvable by the A. K. N. S. –Z. S. scheme must include non-local conserved densities.