Related papers: Asymptotic behavior for a new higher-order nonline…

In this work, the Riemann-Hilbert problem for the 3-component Manakov system is formulated on the basis of the corresponding $4\times 4$ matrix spectral problem. Furthermore, by applying the nonlinear steepest descent techniques to an…

We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering…

Basing on our results [1] on a representation of solutions to the Cauchy problem for multidimensional non-viscous Burgers equation obtained by a method of stochastic perturbation of the associated Langevin system, we deduce an explicit…

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…

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…

We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to0$ as $x\to-\infty$ and $q(x,0)\simeq…

We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula…

We consider the asymptotic behavior of solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity. It is known that the leading term of the asymptotic profile for the solution to this problem is the…

In this paper, we mainly investigate the long-time asymptotic behavior of the solution for the coupled dispersive AB system with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method.Based on the…

We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the derivative nonlinear Schr\"odinger (DNLS) equation: $$iq_t+q_{xx}-iq^2\bar{q}_x+\frac{1}{2}|q|^4{q}=0.$$ with steplike initial data: $q(x,0)=0$ for $x\le 0$ and…

We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1.…

A classification of large-time and finite-time blow-up asymptotics of solutions of the Cauchy problem for higher-order Schr\"odinger equations is performed.

In this work, we investigate the long-time asymptotic behavior of the Wadati-Konno-Ichikawa equation with initial data belonging to Schwartz space at infinity by using the nonlinear steepest descent method of Deift and Zhou for 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…

Following Deift-Zhou's nonlinear steepest descent method, the long-time asymptotic behavior for the Cauchy problem of the 5th order modified Korteweg-de Vries equation is analyzed. Based on the inverse scattering transform, the 5th order…

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 $C(x_1,x_2,v_1,v_2)=\{(x,t)\in\Re^2:x=x_0+vt$…

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work,…

The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate…

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation $$m_{t}=b u_{x}+\frac{1}{2}\left[m\left(|u|^{2}-\left|u_{x}\right|^{2}\right)\right]_{x}-\frac{1}{2} m\left(u \bar{u}_{x}-u_{x}…

In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions.…