We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function qy at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.
Elliptic sine-Gordon Boundary value problems Integrable PDEs
Mathematics Subject Classification
35J60 35J65 35P25
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This research was partially supported by EPSRC grant EP/E022960/1. ASF would like to express his gratitude to the Guggenheim Foundation, USA.
Ashton, A., Fokas, A.S.: Elliptic boundary value problems in convex polygons with low regularity boundary data via the unified method (2012, under review)
ben-Avraham, D., Fokas, A.S.: The modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence. Phys. Rev. E 64, 016114 (2001)
Bona, J.L., Fokas, A.S.: Initial-Boundary-Value problems for linear and integrable nonlinear dispersive partial differential equations. Nonlinearity 21(10), T195–T203 (2008)
Borisov, A.B., Kiseliev, V.V.: Inverse problems for an elliptic sine-Gordon equation with an asymptotic behaviour of the cnoidal type. Inverse Probl. 5, 959–982 (1989)
Boutet de Monvel, A., Fokas, A.S., Shepelsky, D.: The analysis of the global relation for the nonlinear Schrödinger equation on the half-line. Lett. Math. Phys. 65, 199–212 (2003)
Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. American Mathematical Society, Providence (2000)
Dujardin, G.M.: Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals. Proc. R. Soc. Lond. A 465, 3341–3360 (2009)
Flyer, N., Fokas, A.S.: A hybrid analytical numerical method for solving evolution partial differential equations. I. The half-line. Proc. R. Soc. 464, 1823–1849 (2008)
Gutshabash, E.S., Lipovskii, V.D.: Boundary value problem for the two-dimensional elliptic sine-Gordon equation and its applications to the theory of the stationary Josephson effect. J. Math. Sci. 68, 197–201 (1994)
Lenells, J.: Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole. Commun. Math. Phys. 304, 585–635 (2011)
Lenells, J., Fokas, A.S.: An integrable generalisation of the nonlinear Schrödinger equation on the half-line and solitons. Inverse Probl. 25 (2009)
Pelloni, B.: The asymptotic behaviour of the solution of boundary value problems for the sine-Gordon equation on a finite interval. J. Nonlinear Math. Phys. 12(4), 518–529 (2005b)
Smitheman, S.A., Spence, E.A., Fokas, A.S.: A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon. IMA J. Numer. Anal. 30, 1184–1205 (2010)
Spence, E.A., Fokas, A.S.: A new transform method I: domain dependent fundamental solutions and integral representations. Proc. R. Soc. A 466, 2259–2281 (2010a)
Spence, E.A., Fokas, A.S.: A new transform method II: the global relation, and boundary value problems in polar co-ordinates. Proc. R. Soc. A 466, 2283–2307 (2010b)
Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of numerical physics by the method of the inverse scattering problem I. Funct. Anal. Appl. 8, 226–235 (1974)
Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of numerical physics by the method of the inverse scattering problem II. Funct. Anal. Appl. 13, 166–174 (1979)