Search

The inverse scattering transform for the defocusing-defocusing coupled Hirota equations is strictly discussed with non-zero boundary conditions at infinity including non-parallel boundary conditions, specifically referring to the asymptotic polarization vectors. To address the non-analyticity encountered in some of the Jost eigenfunctions, the "adjoint" Lax pair is employed. The inverse problem is formulated as an appropriate matrix Riemann-Hilbert problem. A key difference between non-parallel and parallel boundary conditions lies in the asymptotic behavior of the scattering coefficients, which significantly impacts the normalization of the eigenfunctions and the properties of sectionally meromorphic matrices within the Riemann-Hilbert problem framework. When the asymptotic polarization vectors are non-orthogonal, two distinct methodologies are introduced to convert the Riemann-Hilbert problem into a series of linear algebraic-integral equations. In contrast, when the asymptotic polarization vectors are orthogonal, only one method is feasible. Ultimately, it is demonstrated that pure soliton solutions do not exist in both orthogonal and non orthogonal polarization vector cases. This study provides a comprehensive framework for analyzing the defocusing-defocusing coupled Hirota equations using the inverse scattering transform, offering new insights into the characteristics and solutions of the equations.

The inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions at infinity is thoroughly discussed. We delve into the analytical properties of the Jost eigenfunctions and scrutinize the characteristics of the scattering coefficients. To enhance our investigation of the fundamental eigenfunctions, we have derived additional auxiliary eigenfunctions with the help of the adjoint problem. Two symmetry conditions are studied to constrain the behavior of the eigenfunctions and scattering coefficients. Utilizing these symmetries, we precisely delineate the discrete spectrum and establish the associated symmetries of the scattering data. By framing the inverse problem within the context of the Riemann-Hilbert problem, we develop suitable jump conditions to express the eigenfunctions. Consequently, we deduce the pure soliton solutions from the defocusing-defocusing coupled Hirota equations, and the double-poles solutions are provided explicitly for the first time in this work.

We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(\R)$ initial data. The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painlev\'e region). The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations., Comment: 37 pages, 7 figures; updated version (v2)

Following Deift-Zhou's nonlinear steepest descent method, the long-time asymptotic behavior for the Cauchy problem of the 5th order modified Korteweg-de Vries equation is analyzed. Based on the inverse scattering transform, the 5th order MKdV is transformed to a 2 by 2 oscillatory Riemann-Hilbert problem, then by manipulating the Cauchy operator and reducing the degree of the phase function, the long-time asymptotics of the solution is given in terms of solutions of the parabolic cylinder equation.