Riemann-Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: (1) Lax representation [L,M]=0; (2) construction of fundamental analytic solutions (FAS); (3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) xi+(x,t,lambda)=xi-(x,t,lambda)G(x,t lambda) on a contour Gamma with sewing function G(x,t,lambda); (4) soliton solutions and possible applications. Step 1 involves several assumptions: the choice of the Lie algebra g underlying L, as well as its dependence on the spectral parameter, typically linear or quadratic in lambda. In the present paper, we propose another approach that substantially extends the classes of integrable NLEE. Its first advantage is that one can effectively use any polynomial dependence in both L and M. We use the following steps: (A) Start with canonically normalized RHP with predefined contour Gamma; (B) Specify the x and t dependence of the sewing function defined on Gamma; (C) Introduce convenient parametrization for the solutions xi +/-(x,t,lambda) of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); (D) use Zakharov-Shabat dressing method to derive their soliton solutions. This requires correctly taking into account the symmetries of the RHP. (E) Define the resolvent of the Lax operator and use it to analyze its spectral properties.

Automated summary

The article introduces the standard approach to integrable nonlinear evolution equations (NLEE) and proposes a new method to extend the classes of integrable NLEE. The advantage of this method is that it allows for any polynomial dependence and uses a reduced Riemann-Hilbert problem and Zakharov-Shabat dressing method to solve the equations and derive soliton solutions.