We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case where the underlying isospectral problem has purely discrete spectrum. To this end we introduce a natural coupling problem for entire functions which serves as a replacement for the usual Riemann–Hilbert approach, which does not work for these kind of problems. As a prototypical example...

The complex stress intensity factor and energy release rate are obtained for two semi-infinite interfacial cracks between two bonded dissimilar elastic strips with equal thickness under inplane deformations. During the procedure, by means of conformal mapping technique, the mixed boundary-value problem is reduced to a standard Riemann–Hilbert problem, which is further solved in closed-form. In ...

This paper is concerned with a semi-infinite interfacial crack between two bonded dissimilar elastic strips with equal thickness. Solutions for the complex stress intensity factor (SIF) and energy release rate (ERR) are obtained in closed form under in-plane deformations. During the procedure, the mixed boundary-value problem is reduced by means of the conformal mapping technique to the standar...

This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and the pressure in the air above is constant. It is not supposed that the waves have small amplitude. The problem of existence of such waves is addressed using...

The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann–Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solut...

The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann–Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solut...

In this paper, we obtain the long-time asymptotics of complex mKdV equation via Defit-Zhou method (Non-linear steepest descent method). The Cauchy problem is transformed into corresponding Riemann-Hilbert on basis Lax pair and scattering matrix. After that problems are converted through a decomposition matrix-valued spectral function factorizations jump matrix for problem. Finally, by solving l...