Quantum mechanics

Inverse scattering problem

Riemann surface

Philosophy

Quantum inverse scattering method

Mathematical physics

Korteweg–de Vries equation

Boundary (topology)

Lax pair

Materials science

Boundary value problem

Inverse scattering transform

Scattering

Inverse

Soliton

Simple (philosophy)

Breather

Matrix (chemical analysis)

Physics

Nonlinear system

Riemann hypothesis

Integrable system

Mathematics

Optics

Epistemology

Mathematical analysis

Geometry

Composite material

Inverse problem

Physica D Nonlinear Phenomena

We explore the inverse scattering transforms with matrix Riemann–Hilbert problems for both focusing and defocusing modified Korteweg–de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity systematically. Using a suitable uniformization variable, the direct and inverse scattering problems are proposed on a complex plane instead of a two-sheeted Riemann surface. For the direct scattering problem, the analyticities, symmetries and asymptotic behaviors of the Jost solutions and scattering matrix, and discrete spectra are established. The inverse problems are formulated and solved with the aid of the matrix Riemann-Hilbert problems, and the reconstruction formulas, trace formulas, and theta conditions are also posed. In particular, we present the general solutions for the focusing mKdV equation with NZBCs and both simple and double poles, and for the defocusing mKdV equation with NZBCs and simple poles. Finally, some representative reflectionless potentials are in detail studied to illustrate distinct nonlinear wave structures containing solitons and breathers for both focusing and defocusing mKdV equations with NZBCs.

Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions