Focusing and defocusing mKdV equations with nonzero boundary conditions: inverse scattering transforms and soliton interactions. (English) Zbl 1492.35292
Summary: 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.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 35C08 | Soliton solutions |
| 35P25 | Scattering theory for PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
focusing and defocusing mKdV equations; non-zero boundary conditions; direct and inverse scattering; Riemann-Hilbert problem; multi-soliton solutions; breathersReferences:
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