# A Riemann-Hilbert approach to the Harry-Dym equation on the line

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We consider the initial value problem for the focusing nonlinear Schrï¿½dinger equation with ï¿½step-likeï¿½ initial data: q(x, 0) = 0 for x = 0 and q(x, 0) = Aexp(-2iBx) for x > 0, where A > 0 and B ? R are constants. The paper aims at studying the long-time asymptotics of the solution...

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It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrï¿½dinger equation on the half-line can be expressed through the solution of a 2ï¿½2 matrix Riemannï¿½Hilbert problem. This problem is specified by the spectral functions...

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Two models for a crack in an unbounded two-dimensional micropolar solid are analysed. The first one concerns plane-strain deformation, while the second problem assumes the conditions of antiplane-strain loading. In both cases, one of three modes is uncoupled, while the other two are coupled. For...