sinh-Gordon Equation Research Articles

Ramification of Liouville integrability: Statistical mechanics of the integrable sinh- and sine-Gordon fields

The spectral transform discovered by Gardner, Greene, Kruskal and Miura (GGKM) is a canonical transformation. Functional integration on the classical action offers a direct map between the classical and quantum integrable systems; but in order to compare with the Bethe ansatz method and the quantum inverse method it is necessary to evaluate the functional integrals in terms of spectral data. We report on how this is done for the sinh-Gordon equation and draw comparisons with the similar calculations for the non-linear Schrödinger and sine-Gordon equations.

Sinh-Gordon equation, Painlevé property and Bäcklund transformation

The Painlevé property for partial differential equation proposed by Weiss et al. is studied for the sinh-Gordon equation. We performed a singular point analysis to show that the Painlevé property is a valuable analytical test for integrability and we found in a remarkably straightforward manner the Bäcklund transformation of the sinh-Gordon equation.

Soliton Connection of the sinh-Gordon Equation

It is demonstrated that the sinh-Gordon equation can be written as covariant exterior derivatives of Lie algebra valued differential forms and, moreover, that these nonlinear differential equations represent a completely integrable system.

Two explicit unconnected sets of elliptic solutions parametrized by an arbitrary function for the two-dimensional Toda chain A1(1)

Two sets of explicit elliptic solutions, parametrized by an arbitrary holomorphic function are presented for the two-dimensional Toda chain A 1 (1), obtained by means of reduction from O(3) and O(2,1) σ-models, having elliptic solutions, parametrized by an arbitrary function. A particular case — the sinh-Gordon equation is considered.

Comments on Periodic Waves and Solitons

Elementary arguments are given for various results in wave propagation. The first part concerns the representation of periodic waves as sums of solitons. These are given for the Korteweg—deVries, various modified Korteweg—deVries and Boussinesq equations. The analogy with Parker's solution for Burgers' equation, which uses a sum of shocks, is noted. The second part provides elementary proofs for some of the existing results on multiphase solutions to the Korteweg—deVries and Sinh-Gordon equations.

The elliptic solution of the sinh-Gordon equation

The elliptic solution of the two-dimensional Toda chain with the Cartan matrix of the Kac-Moody algebra A (1) 1, parametrized by an arbitrary (anti) holomorphic function is obtained, as well as its particular realization - the sinh-Gordon equation - with the use of the generalized Pohmeyer transformation and the generalization of the elliptic solution of the CP 1 model.

Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint

A summary of the spectral theory for quasiperiodic sine- and sinh-Gordon equations is given. Analogies with whole-line solitons and scattering theory motivates the discussion. The relation between the ingredients in the inverse spectral solution of the periodic sine-Gordon equation and physical characteristics of sine-Gordon waves is emphasized. The explicit topics covered are summarized in the table of contents in the Introduction.

The Maxwell and Proca equations in spaces with torsion

Nonlinear equations are derived for mass as well as massless vector fields in a Riemann space, where nonlinearity is induced by the interaction of these fields with individual irreducible parts of the torsion tensor. It is shown that in the simplest cases these equations are exactly solvable for an electromagnetic field, while a mass field is described by a number of interesting equations, in particular the sinh-Gordon equation, which has soliton solutions.

Connection among Equations for the Classical Heisenberg Ferromagnet, the Ernst Equation for the Axisymmetric Gravitational Field and the Yang Equations for the Self-Dual Gauge Field

A study is made on the connection among stationary equations for the classical continuous isotropic Heisenberg model (CCIHM), the Ernst equation for the axisymmetric gravitational field (ASGF) and the Yang equations for the self-dual SU(2) gauge field (SDGF). This is done by reducing the Yang equations to a four-dimensional (4d) version of the Ernst equation and by parametrizing the Ernst potential in terms of variables θ and φ analogous to two angles of rotation of spins in the CCIHM. Both of the Ernst equation and the Yang equations are thereby reduced to a pair of equaions for θ and φ quite similar to those for the CCIHM, the latter reducing to the former by taking θ → iθ. This shows one-to-one correspondence of solutions to the field equations between the CCIHM and the SDGF or the ASGF, though spatial dimensionality, symmetry and boundary condition of physical significance may be different for these different cases. By paying attention to particular solutions for which φ satisfies the Laplace equation, three types of solutions are obtained for the SDGF by using analogy with hydrodynamics, one of which is also applicable to the ASGF. For the first type new multi-vortex or multi-instanton solutions similar to vortex solutions for the two-dimensional (2d) CCIHM are obtained for the 4d SDGF. The second type is axially symmetric solutions for the three-dimensional (3d) SDGF and the ASGF written in terms of the Painlevé transcendents of the third kind. The third type is uiform-flow solutions for the 3d SDGF given as solutions to the 2d static sinh-Gordon equation.

The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras

AbstractWe associate to each complex simple Lie algebra g a hierarchy of evolution equations; in the simplest case g = sl(2) they are the modified KdV equations. These new equations are related to the two-dimensional Toda lattice equations associated with g in the same way that the modified KdV equations are related to the sinh-Gordon equation.

Singular solitons: An example of a Sinh-Gordon equation

The application of the inverse scattering transform for a description of singular solutions of nonlinear equations is considered, using an example of a Sinh-Gordon equation. The singular solitons are constructed. The soliton lines of singularity are interpreted as world lines of a Poincare-invariant dynamical system of interacting spin particles.

The sine‐gordon and sinh‐gordon equations on the circle

The sine‐gordon and sinh‐gordon equations on the circle

A string-like self-dual solution of Yang-Mills theory

We attempt to obtain Nielsen-Olesen strings in pure Yang-Mills theory without Higgs scalars. This is inspired by recent work interpreting the Prasad-Sommerfield monopole as a static self-dual Yang-Mills solution in which A 4 a plays the role of the Higgs field. In similar fashion, we make A 3 a A 4 a serve as the two required isovector fields in an ansatz independent of x 3 and x 4. The condition of self-duality results in a single Painlevé equation of the third kind (or equivalently, a radial sinh-Gordon equation in 2 + 0 dimensions), the solution of which determines A μ a . We make use of the extensive analysis of the former equation carried out by Wu, McCoy and collaborators in the context of the scaling limit of the two-dimensional Ising model. Their simplest solution yields a flux value of −2 π/ e just as in the Nielsen-Olesen model and flux is quantized in multiples of this unit. The string tension (action per unit time per unit distance) diverges as r −2 (In r) −2 as r → 0 for the same Wu-McCoy solution.

Integrability of the classical $$[\bar \psi _i \psi _i ]_2^2 $$ and $$[\bar \psi _i \psi _i ]_2^2 - [\bar \psi _i \gamma _5 \psi _i ]_2^2 $$ interactions

We study the interaction ofN classical two-dimensional massless Fermi fields through the symmetric couplings\([\bar \psi _i \psi _i ]^2 \) or\([\bar \psi _i \psi _i ]^2 - [\bar \psi _i \gamma _5 \psi _i ]^2 \). We explicitly show complete integrability in the casesN=1, 2, using the inverse scattering method. The fields occuring in the associated linear eigenvalue problem and evolution equation are simply related to the fundamental fields Ψi that satisfy the original non-linear equations. ForN>2, calculations become very involved, but there is no doubt that the system remains completely integrable, reducing to appropriate generalizations of the sine- and sinh-Gordon equation, a situation analogous to Pohlmeyer's discussion in a somewhat similar problem: the two-dimensional non-linear σ-model. Finally, all the explicit analytic solutions that we have worked out in the present framework are identical to those found by Dashen et al., and Shei, in a semiclassical treatment of the fully quantum mechanical version of these models. This leads us to conjecture that the quantum theory also shares most of the features of completely integrable systems, like the massive Thirring model.

AnN-soliton solution of a nonlinear optics equation derived by a general inverse method

Two recent papers have considerably extended the use of the (, inverse scattering method in solving the init ial-value problem for some nonlinear par t ia l differential equations. Firs t ly , ZAKnAaOV and SHAB*T (t) successfully used the method, originally suggested by GAnDNEI~ et al. (~), to solve the init ial-value problem for the self-focussing equation of nonlinear optics. Secondly, ABLOWITZ et al. (3) extended this approach by treating a broad class of equations as a single problem. In this way they solved the init ial-value problem for the Korteweg-de Vries and modified Korteweg-de Vries equations, the sine-Gordon and sinh-Gordon equations, and the Benney-Newell equation. The first two equations in this group were solved separa.tely in two previous publications (a.5). An important application of the inverse method in solving an initial-vahie problem follows from the fact tha t the discrete par t of the eigenvalue spectrum corresponds to soliton solutions of the equation. These solutions are of considerable interest in ultrashort-optical-pulse propagation (~) and many other branches of physics (7). In this letter, we solve the initiM-value problem for a set of equations occurring in nonlinear optics, called the reduced Maxwell-Bloch (RMB) equations(~). From this solution we derive, by ~ nontr ivial transformation, an N-soliton solution to the RMB equations which confirms m ,V-soliton solution proposed by the authors of this article (s). The most impor tant point in our present approach to this problem is tha t we can fit the RMB equations into the general scheme proposed by ABLOWI~Z et al. (a). This means tha t we not only considerably simplify our previous proof (9) for all ~V-soliton

Nonlinear-Evolution Equations of Physical Significance

We present the inverse scattering method which provides a means of solution of the initial-value problem for a broad class of nonlinear evolution equations. Special cases include the sine-Gordon equation, the sinh-Gordon equation, the Benney-Newell equation, the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, and generalizations.