Related papers: Long-Time Asymptotics for the Camassa-Holm Equatio…
In this paper, we consider the initial value problem for the complex short pulse equation with a Wadati-Konno-Ichikawa type Lax pair. We show that the solution to the initial value problem has a parametric expression in terms of the…
In this paper, we obtain the long-time asymptotics of complex mKdV equation via Defit-Zhou method (Non-linear steepest descent method). The Cauchy problem of complex mKdV equation is transformed into the corresponding Riemann-Hilbert…
In this work, we study the Cauchy problem of the Elastic Beam equation with initial value in weighted Sobolev space $H^{1,1}(\mathbb{R})$ via the $\bar{\partial}$-steepset descent method. Begin with the Lax pair of the Elastic Beam…
We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the…
In this work, we consider the Cauchy problem for the defocusing Hirota equation with a nonzero background \begin{align} \begin{cases} iq_{t}+\alpha\left[q_{xx}-2\left(\left\vert…
In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long…
We prove large time asymptotics for solutions of the KP I equation with small initial data. Our assumptions on the initial data rule out lump solutions but give a precise description of the radiation field at large times. Our analysis uses…
We investigate the long-time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with nonzero boundary conditions in different regions \begin{align*} &m_{t}+\left((u^2-u_x^2)m\right)_{x}=0,~~ m=u-u_{xx},…
We consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and…
This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection…
The Sasa-Satsuma equation with $3 \times 3 $ Lax representation is one of the integrable extensions of the nonlinear Schr\"{o}dinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying…
The long-time asymptotics of solutions of the Cauchy problem for the heat equation are constructed in the case when the initial function at infinity has power asymptotics.
This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by…
We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for…
I derive a Fredholm determinant for the thermal correlator of vertex operators with a conformally invariant density matrix and nonlinear time evolution. Using the method of nonlinear steepest descent I find the long time asymptotics at…
We consider the initial value problem for a new two-component Sasa-Satsuma equation associated with the fourth-order Lax pair with decaying initial data on the line. By utilizing the spectral analysis, the solution of the new two-component…
In this paper, we mainly focus on the Cauchy problem of an integrable nonlocal Hirota equation with initial value in weighted Sobolev space. Through the spectral analysis of Lax pairs, we successfully transform the Cauchy problem of the…
In this paper we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations,…
The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization…
We prove a general asymptotic decay lemma which is applicable in various contexts. As an example, the general theorem is shown to give lower growth estimates for entire and exterior solutions of the minimal surface equation.