Nonzero Boundary Question To Toda Equation

Zero boundary condition for the Toda equation is not suitable. The article extends the Riemann-Hilbert approach to nonlinear integrable Toda equation with nonzero boundary condition. Inverse scattering transform of this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary condition. To get new sectionally analytic functions without null point on boundary curves, we consider the reflectionless potentials. Through regularization of the Riemann-Hilbert problem, we establish a relection between the solutions of Toda equation and soliton matix. Finally, solitonic solutions to the Toda equation are given.