Small-amplitude solutions of the sine-Gordon equation on an interval under dirichlet or Neumann boundary conditions

Summary We give a complete classification of the small-amplitude finite-gap solutions of the sine-Gordon (SG) equation on an interval under Dirichlet or Neumann boundary conditions. Our classification is based on an analysis of the finite-gap solutions of the boundary problems for the SG equation by means of the Schottky uniformization approach.On leave from IPPI, Moscow, Russia     

Finitary automorphisms of groups

We introduce the so-called multiscale limit for spectral curves associated with real finite-gap sine-Gordon solutions. This technique allows us to solve the old problem of calculating the density of the topological charge for real finite-gap sine-Gordon solutions directly from the θ-functional formulae.     

Finite-gap solutions of the Davey-Stewartson equations

Summary We describe the finite-gap solutions of different modifications of the Davey-Stewartson (DS) equations. The restrictions on the spectral data which give us solutions of the real forms DS1 and DS2⁺ of DS are the same as those in the case of KP1 and KP2 of the Kadomtsev-Petviashvily equation. But for DS2⁻ the restrictions that we regard have no analogues in other integrable systems. We describe also the restrictions that provide regularity of those solutions for DS1 and DS2^±. The finite-gap solutions include rational and soliton solutions. We give some classes of those solutions. The well-known dromions for DS1 are examples of that kind.     

The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Several results including integral representation of solutions and Hermite– Krichever Ansatz on Heun’s equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin’s solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich–Verdier potential and additional apparent singularities of exponents − 1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schr?dinger operator on our potential.     

Finite-gap elliptic solutions of the KdV equation

A method is proposed for constructing finite-gap elliptic inx or/and int solutions of the Korteweg-de Vries equation. Dynamics of poles for two-gap elliptic solutions of the KdV equation are considered. Numerous examples of new elliptic solutions of the KdV equation are given.Dedicated to the memory of J.-L. Verdier     

Temporal asymptotics for soliton equations in problems with step initial conditions

In this survey, we present modern approaches to the construction and justification of large time asymptotics for solutions of main soliton equations with step-like initial condition whose boundary conditions as x ± are finite-gap, quasi-periodic solutions. The principal term of the asymptotic is also a finite-gap, quasi-periodic solution whose phase vectors are modulated with respect to the slow space-like variable. The Whitham equations describing this modulation are studied in detail. For the KdV equations, we construct and justify the principal term of the asymptotic for arbitrary finite-gap boundary conditions. By examining the sine-Gordon equation, we study the case of boundary conditions with complex-valued, self-conjugated quasi-periods. We prove the existence and uniqueness theorems in the case of the complex-valued group velocities that appear in the Whitham equations in the case considered. We present a complete picture (uniform in x) of the Whitham deformation for the case of 1-gap boundary conditions in the sine-Gordon equation.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003.     

The nonlinear Klein-Gordon equation on an interval as a perturbed Sine-Gordon equation

We treat the nonlinear Klein-Gordon (NKG) equation as the Sine-Gordon (SG) equation, perturbed by a higher order term. It is proved that most small-amplitude finite-gap solutions of the SG equation, which satisfy either Dirichlet or Neumann boundary conditions, persist in the NKG equation and jointly form partial central manifolds, which are “Lipschitz manifolds with holes”. Our proof is based on an analysis of the finite-gap solutions of the boundary problems for SG equation by means of the Schottky uniformization approach, and an application of an infinite-dimensional KAM-theory. The first author was supported by the Alexander von Humbold Foundation and the Sonder-forschungsbereich 288.     

Landau-Lifshitz equation,uniaxial anisotropy case: Theory of exact solutions

Using the inverse scattering method, we study the XXZ Landau-Lifshitz equation well-known in the theory of ferromagnetism. We construct all elementary soliton-type excitations and study their interaction. We also obtain finite-gap solutions (in terms of theta functions) and select the real solutions among them.     

The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation

We study a system of particles on a Riemann surface with a puncture. This system describes the behavior of zeros of finite-gap solutions of the Schrödinger equation corresponding to a degenerate hyperelliptic curve. We show that this system is Hamiltonian and integrable by constructing action-angle type coordinates.     

Finite-gap attractors and transition processes of the shock-wave type in integrable systems

An attractor interpretation of finite-gap solutions in integrable KdV-type systems is suggested. Certain shock-wave problems in these systems are discussed. Bibliography: 25 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 25–36. Translated by R. F. Bikbaev.     

The AKNS Hierarchy and Finite-Gap Schrödinger Potentials

We consider the AKNS hierarchy and find the necessary and sufficient conditions for functions p and q to become solutions of the AKNS hierarchy. Using the functions p and q, we construct finite-gap Schrödinger potentials.     

On a class of elliptic solutions of the Boussinesq equations

An ansatz is considered for Krichever curves associated with doubly periodic finite-gap solutions of the Boussinesq equation expressed in terms of equi-anharmonic elliptic Weierstrass &-functions. Some examples are given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 347–356, December, 1996.     

The Elliptic Gaudin System with Spin

We study the algebraic–geometric structure of the elliptic Gaudin two-puncture model previously obtained. We identify this system with the system of pole dynamics of finite-gap solutions of the matrix Davey–Stewartson equation. We also obtain the action–angle variables and construct explicit solutions of this system in terms of theta functions. We discuss the geometry of degenerations of this system.     

Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.     

Some algebraic examples of Frobenius manifolds

Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity equations and the nonsemisimple Frobenius manifolds associated with them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007.     

The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker-Akhiezer Function

We obtain a simple sufficient condition for the solvability of the Riemann factorization problem for matrix-valued functions on a circle. This condition is based on the symmetry principle. As an application, we consider nonlinear evolution equations that can be obtained by a unitary reduction from the zero-curvature equations connecting a linear function of the spectral parameter z and a polynomial of z. We consider solutions obtained by dressing the zero solution with a function holomorphic at infinity. We show that all such solutions are meromorphic functions on ℂ _xt ² without singularities on ℝ _xt ² . This class of solutions contains all generic finite-gap solutions and many rapidly decreasing solutions but is not exhausted by them. Any solution of this class, regarded as a function of x for almost every fixed t ∈ ℂ, is a potential with a convergent Baker-Akhiezer function for the corresponding matrix-valued differential operator of the first order.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 453–471, September, 2005.     

Cylindrical Kadomtsev-Petviashvili equation: Old and new results

We review results on the cylindrical Kadomtsev-Petviashvili (CKP) equation, also known as the Johnson equation. The presentation is based on our results. In particular, we show that the Lax pairs corresponding to the KP and the CKP equations are gauge equivalent. We also describe some important classes of solutions obtained using the Darboux transformation approach. We present plots of exact solutions of the CKP equation including finite-gap solutions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 304–320, August, 2007.     

One-Point Commuting Difference Operators of Rank One and Their Relation with Finite-Gap Schrödinger Operators

One-point commuting difference operators of rank one in the case of hyperelliptic spectral curves are studied. A relationship between such operators and one-dimensional finite-gap Schrödinger operators is investigated. In particular, a discretization of finite-gap Lamé operators is obtained.     

Reduction of the dressing chain of the Schrödinger operator

We reduce the problem of constructing real finite-gap solutions of the focusing modified Korteweg-de Vries equation, to the dressing chain of the Schrödinger operator. We show that the Schrödinger operator spectral curve corresponding to such a solution is real. We give some restrictions on the initial data for the chain that lead to such solutions. We also consider a soliton, reduction. We obtain compact representations for the multisoliton and breather solutions of the modified Korteweg-de Vries equations; these representations can be useful in developing the perturbation theory for various applied problems.