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     

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.     

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.     

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     

Exact solutions to the double Sine-Gordon equation

The double Sine-Gordon equation (DSG) with arbitrary constant coefficients is studied by F-expansion method, which can be thought of as an over-all generalization of the Jacobi elliptic function expansion since F here stands for every one of the Jacobi elliptic functions (even other functions). We first derive three kinds of the generic solutions of the DSG as well as the generic solutions of the Sine-Gordon equation (SG), then in terms of Appendix A, many exact periodic wave solutions, solitary wave solutions and trigonometric function solutions of the DSG are separated from its generic solutions. The corresponding results of the SG, which is a special case of the DSG, can also be obtained.     

The Abreu equation with degenerated boundary conditions

The Abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds. We study the Dirichlet problem of the Abreu equation with degenerated boundary conditions. The solutions provide the Kähler metrics of constant scalar curvature on the complex torus.     

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.     

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 boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation

This article describes a numerical method based on the boundary integral equation and dual reciprocity method for solving the one‐dimensional Sine‐Gordon (SG) equation. The time derivative is approximated by the time‐stepping method and a predictor–corrector scheme is employed to deal with the nonlinearity which appears in the problem. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. In addition, the conservation of energy in SG equation is investigated. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

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.     

Eigenfunctions of the Dirichlet and Neumann boundary value problems for the elliptic sinh-Gordon equation on a rectangle

The Dirichlet and Neumann zero boundary value problems on a rectangle for the equation u + sinhu=0 are considered. Exact solutions are constructed by means of finite-gap integration theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 32–36, 1989.     

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 wave equation on asymptotically de Sitter-like spaces

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X^○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y_±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y₊ to Y₋.     

Integrable boundary-value problem for the Volterra chain on the half-axis

We study quasiperiodic (finite-gap) solutions of the Volterra chain satisfying an integrable boundary condition on the semiaxis. From the set of general finite-gap solutions, only those corresponding to the boundary-value problem are singled out, the relevant condition being expressed as a system of algebraic equations.     

Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation u_t = u_xx + f(u),   0 < u(x,t) < 1, (x,t) ∈ ℝ², where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.     

Finding the unknown boundary in the initial boundary-value problem for the heat equation

The article considers the determination of the boundary of a two-dimensional region in which an initial boundary-value problem for the heat equation is defined, given the solution of the problem for all time instants at some points of the region. The direct problem is reduced to an integral equation, and numerical solutions of the inverse problem are obtained for the case when the boundary is an ellipse. We investigate the sensitivity of the observed variables to the location (relative to the boundary) of the point where the right-hand side of the equation is specified. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 18–24.     

Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds

We study the C^1,1 and Lipschitz regularity of the solutions of the degenerate complex Monge-Ampère equation on compact K?hler manifolds. In particular, in view of the local regularity for the complex Monge-Ampère equation, the obtained C^1,1 regularity is a generalization of the Yau theorem which deals with the nondegenerate case. Received: 18 April 2002 / Published online: 24 February 2003 Partially supported by KBN Grant #2 P03A 028 19     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.