arXiv subjects

Ruihong Ma

Publications and source records attributed to Ruihong Ma.

5 recordsLinked to original sources

arXiv2026

Painlev\'e XXXIV Asymptotics for the Focusing mKdV Equation with Finite-Genus Background and Discrete Spectrum

We investigate the Cauchy problem for the focusing modified Korteweg--de Vries (mKdV) equation with finite-genus algebro-geometric quasi-periodic initial data. By applying the nonlinear steepest-descent method of Deift--Zhou to the associated Riemann--Hilbert (RH) problem, we derive the long-time asymptotics of the solution in the critical regime where complex stationary phase points coalesce with the endpoints of the finite-genus branch cuts. The collision is resolved via a local Painlev\'{e} XXXIV parametrix, and the discrete spectrum (breathers) is incorporated into the analysis. The resulting expansion is valid uniformly up to an error of order $\mathcal{O}(t^{-1/2})$. In this critical region, the leading-order term comprises the finite-genus algebro-geometric background together with breathers, whose parameters are slowly modulated by the background solution.

math-ph↗

arXiv2024

Soliton Shielding of the focusing modified KdV equation

We consider soliton gas solutions of the modified Korteweg-de Vries (mKdV) equation, where the point spectrum of the condensate is located within a bounded domain in the upper half-plane. We first demonstrate that when the domain is a quadrature and the soliton density is an analytic function, the corresponding deterministic soliton gas coincides with a finite number of solitons, which we call this effect soliton shielding. When the domain is an ellipse and the soliton density is analytic, the corresponding deterministic soliton gas reduces the spectral data to the segment joining the foci. The initial datum of this Cauchy problem is asymptotically step-like oscillatory, described by a periodic elliptic function as $ x \to +\infty $, and it vanishes exponentially fast as $ x \to -\infty $.

math-ph↗

arXiv2024

Soliton resolution and asymptotic stability of $N$-loop-soliton solutions for the Ostrovsky-Vakhnenko equation

The Ostrovsky-Vakhnenko (OV) equation \begin{align*} &u_{txx}-3κu_x+3u_xu_{xx}+uu_{xxx}=0 \end{align*} is a short wave model of the well-known Degasperis-Procesi equation and admits a $3\times 3$ matrix Lax pair. In this paper, we study the soliton resolution and asymptotic stability of $N$-loop soliton solutions for the OV equation with Schwartz initial data that supports soliton solutions. It is shown that the solution of the Cauchy problem can be characterized via a $3\times 3$ matrix Riemann-Hilbert (RH) problem in a new scale. Further by deforming the RH problem into solvable models with $\bar\partial$-steepest descent method, we obtain the soliton resolution to the OV equation in two space-time regions $x/t>0$ and $x/t<0$. This result also implies that $N$-loop soliton solutions of the OV equation are asymptotically stable.

math-ph↗

arXiv2023

The asymptotic stability of solitons in the focusing Hirota equation on the line

In this paper, the $\overline\partial$-steepest descent method and Bäcklund transformation are used to study the asymptotic stability of solitons to the Cauchy problem of focusing Hirota equation. The solution of the RH problem is further decomposed into pure radiation solution and solitons solution obtained by using $\overline\partial$-techniques and Bäcklund transformation respectively. As a directly consequence, the asymptotic stability of solitons for the Hirota equation is obtained.

math.AP↗

arXiv2019

Long time asymptotics behavior of the focusing nonlinear Kundu-Eckhaus equation

We study the Cauchy problem for the focusing nonlinear Kundu-Eckhaus equation and construct long time asymptotic expansion of its solution in fixed space-time cone with $C(x_1,x_2,v_1,v_2)=\{(x,t)\in\Re^2:x=x_0+vt$ $x_0\in[x_1,x_2],v\in[v_1,v_2] \}$. By using the inverse scattering transform, Riemann-Hilbert approach and $\overline\partial$ steepest descent method we obtain the lone time asymptotic behavior of the solution, at the same time we obtain the solitons in the cone compare with the all N-soliton the residual error up to order $\mathcal{O}(t^{-3/4})$.

nlin.SI↗