Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2109.02901

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

Xiao-Fan Zhang·Shou-Fu Tian·Jin-Jie Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We extend the Riemann-Hilbert (RH) method to study the inverse scattering transformation and high-order pole solutions of the focusing and defocusing nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with nonzero boundary conditions (NZBCs) at infinity and successfully find its multiple soliton solutions with one high-order pole and multiple high-order poles. By introducing the generalized residue formula, we overcome the difficulty caused by calculating the residue conditions corresponding to the higher-order poles. In accordance with the Laurent series of reflection coefficient and oscillation term, the determinant formula of the high-order pole solution with NZBCs is established. Finally, combined with specific parameters, the dynamic propagation behaviors of the high-order pole solutions are further analyzed and some very interesting phenomena are obtained, including kink solution, anti kink solution, rational solution and breathing-soliton solution.

Explore related subjects

nlin.SImath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xiao-Fan Zhang, Shou-Fu Tian, Jin-Jie Yang. 2021-09-07. 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. https://arxiv.org/abs/2109.02901

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Superintegrability of discrete-time rational Ruijsenaars-Schneider model and deformed polynomial symmetry algebras

We explicitly construct the additional integrals of motion, ensuring maximal superintegrability of the discrete-time rational Ruijsenaars-Schneider model. Using them, we investigate the algebraic aspects of superintegrability in both continuous- and discrete-time settings. In particular, we determine the complete structures of the polynomial symmetry algebras associated with both the rational Ruijsenaars-Schneider model and its discretization. We demonstrate that discretization leads to a nontrivial deformation of the continuous symmetry algebra with respect to the discretization parameter, thereby extending recent analogous results from the rational Calogero-Moser system to its relativistic generalization.

nlin.SI↗
arXiv2026

Large-space and Large-time Asymptotics for the Focusing Nonlinear Schrödinger Soliton Gas

We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schrödinger equation. The soliton gas is constructed as the continuum limit of pure $N$-soliton solutions as $N\to\infty$, with the discrete spectrum confined to two segments $Σ_1$ and $Σ_2$. In particular, our framework does not require the discrete spectrum to be confined to the imaginary axis. By combining the nonlinear steepest descent method with an appropriate $g$-function mechanism, we show that, as $x\to-\infty$, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime $t\to+\infty$, we assume that the endpoint $F$ lies on the trajectory of $H(ξ)$ with $ξ=\frac{x}{2t}\in(-E_1-\sqrt{2}E_2,-E_1)$, namely, $F=H(\hatξ)$, $\hatξ\in (-E_1-\sqrt{2}E_2,-E_1)$. Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable $ξ=\frac{x}{2t}$. More precisely, there exist an exponentially decaying region $ξ\in(-E_1,+\infty)$, a modulated elliptic-wave region $ξ\in(\hatξ,-E_1)$, and an unmodulated elliptic-wave region $ξ\in(-\infty,\hatξ)$.

nlin.SI↗
arXiv2026

On singular solitons of the KP equation and the Go-diagrams

It has been proven that real and regular soliton solutions of the KP equation are classified in terms of the totally nonnegative Grassmannian. It is well known that vertex operators can be used to construct soliton solutions. In this paper, we consider several regular soliton solutions and study their combinations through products of vertex operators. In general, the resulting solutions become singular. Totally nonnegative elements are parametrized by the Le-diagrams introduced by Postnikov. We show that the resulting singular solutions can be parametrized by Go-diagrams, which extend Le-diagrams and arise in the Deodhar decomposition of the Grassmannian.

nlin.SI↗