Inverse scattering transforms for the focusing and defocusing mKdV equations with nonzero boundary conditions

Abstract

We explore systematically a rigorous theory of 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. 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 Riemann-Hilbert problems, and the reconstruction formulae, trace formulae, and theta conditions are also found. 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 wave structures for both focusing and defocusing mKdV equations with NZBCs.

Inverse scattering transforms and N-double-pole solutions for the derivative NLS equation with zero/non-zero boundary conditions

“…Recently, the ISTs of integrable nonlinear systems with NZBCs have attracted more and more attention [23][24][25][26][27][28][29].…”

“…Moreover, no trace formulae or theta conditions were researched, the inverse problems were not formulated by means of Riemann-Hilbert problems. A natural problem is whether explicit multi-pole solutions can be found for the DNSL equation with ZBC/NZBCs by the approximate IST based on the matrix Riemann-Hilbert problems [23][24][25][26][27][28][29].…”

“…In the present paper, we present the ISTs for the DNLS equation with ZBC/NZBCs based on the matrix Riemann-Hilbert problems (RHPs), and present their novel double-pole solutions by solving the corresponding RHPs. It should pointed out the DNLS equation is associated with the modified Zakharov-Shabat eigenvalue problem, not the usual Zakharov-Shabat eigenvalue problem related to the NLS-type equations [21][22][23][24][25][26][27][28][29] such that the discrete spectrum and solving Riemann-Hilbert problems are more complicated.…”

Inverse scattering transform and multiple high-order pole solutions for the nonlocal focusing and defocusing modified Korteweg-de Vries equation with the nonzero boundary conditions

“…Then, the IST method of focusing and defocusing mKdV equation and the derivative NLS equation with NZBCs were studied at infinity in 2019 [36,37]. Inspired by Yan's work, we employ the RH method to study the integrable real nonlocal mKdV equation [38] q t (x, t) − 6σq(x, t)q(−x, −t)q x (x, t) + q xxx (x, t) = 0, (…”

Riemann-Hilbert approach to the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions

“…After that, it is more and more popular to use the IST to study the NLS equations. Although the IST is extended to study the NLS equations with NZBCs [29]- [52], there are very few work to study the case of variable coefficients.…”