Author

# Zechuan Zhang

Bio: Zechuan Zhang is an academic researcher from Fudan University. The author has contributed to research in topics: Type (model theory) & Method of steepest descent. The author has an hindex of 1, co-authored 3 publications receiving 4 citations.

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Fudan University

^{1}TL;DR: In this article, the authors investigated the long time asymptotics for the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data in different solitonic regions.

Abstract: We investigate the long time asymptotics for the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data in different solitonic regions \begin{align*}
&q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \quad (x,t)\in\mathbb{R}\times \mathbb{R}^{+},
&q(x,0)=q_{0}(x), \quad \lim_{x\rightarrow\pm\infty}q_{0}(x)=\pm 1, \end{align*} where $q_0\mp 1\in H^{4,4}(\mathbb{R})$.Based on the spectral analysis of the Lax pair, we express the solution of the mKdV equation in terms of a Riemann-Hilbert problem. In our previous article, we have obtained long time asymptotics and soliton resolutions for the mKdV equation in the solitonic region $\xi\in(-6,-2)$ with $\xi=\frac{x}{t}$.In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for the solitonic region $\xi\in(-\varpi,-6)\cup(-2,\varpi)$ with $ 6 < \varpi<\infty$ being an arbitrary constant.For $-\varpi<\xi<-6$, there exist four stationary phase points on jump contour, and the asymptotic approximations can be characterized with an $N$-soliton on discrete spectrums and a leading order term $\mathcal{O}(t^{-1/2})$ on continuous spectrum up to a residual error order $\mathcal{O}(t^{-3/4})$. For $-2<\xi<\varpi$, the leading term of asymptotic expansion is described by the soliton solution and the error order $\mathcal{O}(t^{-1})$ comes from a $\bar{\partial}$-problem. Additionally, asymptotic stability can be obtained.

8 citations

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TL;DR: In this article, the authors considered the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation for finite density type initial data.

Abstract: We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation for finite density type initial data. With the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method, we extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic region of space-time, $|x/t+4|<2$, and we give bounds for the error which decay as $t\rightarrow\infty$ for a general class of initial data whose difference from the non-vanishing background possesses a fixed number of finite moments. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for mKdV equation.

1 citations

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Fudan University

^{1}TL;DR: In this paper, the authors considered the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with finite density type initial data.

Abstract: We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with finite density type initial data. With the $\bar{\partial}$ generalization of the nonlinear steepest descent method of Deift and Zhou, we extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic space-time region $|x/t+4|<2$, and we give bounds for the error which decay as $t\rightarrow\infty$ for a general class of initial data whose difference from the non-vanishing background possesses a fixed number of finite moments. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of $N$-soliton solutions for mKdV equation with finite density type initial data.

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04 Apr 2022

TL;DR: In this paper , the authors investigated the long time asymptotics for the Cauchy problem of the defocusing modiﬁed Kortweg-de Vries (mKdV) equation with step-like initial data.

Abstract: We investigate the long time asymptotics for the Cauchy problem of the defocusing modiﬁed Kortweg-de Vries (mKdV) equation with step-like initial data, i.e., q 0 ( x ) = q ( x, t = 0) = C L , as x < 0 and q 0 ( x ) = C R as x > 0, where C L > C R > 0 are arbitrary positive real numbers. We ﬁrstly develop the direct scattering theory to establish the Riemann-Hilbert (RH) problem associated with step-like initial data. Then by introducing the related g function in diﬀerent space-time regions and using the steepest descent analysis, we deform the original matrix valued RH problem to explicitly solving models. Finally we obtain the diﬀerent long-time asymptotic behavior of the solution of the Cauchy problem for defocusing mKdV equation in four diﬀerent space-time regions R ξ,I , R ξ,II , R ξ,III and R ξ,IV in the half-plane, where R ξ,I and R ξ,IV are far left ﬁeld regions; and R ξ,II and R ξ,III are rarefaction wave regions.

2 citations

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TL;DR: In this paper , the authors studied the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions.

Abstract: We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions $$\begin{aligned}&q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, \\&\quad q(x,0)=q_{0}(x),\ \ \lim _{x\rightarrow \pm \infty } q_{0}(x)=q_{\pm }, \end{aligned}$$ where $$|q_{\pm }|=1$$ and $$q_{+}=\delta q_{-}$$ , $$\sigma \delta =-1$$ . In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $$-6<\xi <6$$ with $$\xi =\frac{x}{t}$$ . In this paper, we give the asymptotic expansion of the solution q(x, t) for other solitonic regions $$\xi <-6$$ and $$\xi >6$$ . Based on the Riemann–Hilbert formulation of the Cauchy problem, further using the $${\bar{\partial }}$$ steepest descent method, we derive different long time asymptotic expansions of the solution q(x, t) in above two different space-time solitonic regions. In the region $$\xi <-6$$ , phase function $$\theta (z)$$ has four stationary phase points on the $${\mathbb {R}}$$ . Correspondingly, q(x, t) can be characterized with an $${\mathcal {N}}(\Lambda )$$ -soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function $$\textrm{Im}
u (\zeta _i)$$ . In the region $$\xi >6$$ , phase function $$\theta (z)$$ has four stationary phase points on $$i{\mathbb {R}}$$ , the corresponding asymptotic approximations can be characterized with an $${\mathcal {N}}(\Lambda )$$ -soliton with diverse residual error order $${\mathcal {O}}(t^{-1})$$ .

1 citations

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TL;DR: In this article, the authors considered the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation for finite density type initial data.

Abstract: We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation for finite density type initial data. With the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method, we extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic region of space-time, $|x/t+4|<2$, and we give bounds for the error which decay as $t\rightarrow\infty$ for a general class of initial data whose difference from the non-vanishing background possesses a fixed number of finite moments. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for mKdV equation.

1 citations

15 Aug 2022

TL;DR: In this paper , the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions where κ is a positive constant was studied and a Riemann-Hilbert problem corresponding to the original problem was constructed to give the solution of the CH equation with the initial boundary value condition.

Abstract: . In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions where κ is a positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the CH equation with the initial boundary value condition. Furthermore, by developing the ¯ ∂ -generalization of Deift-Zhou nonlinear steepest descent method, diﬀerent long-time asymptotic expansions of the solution q ( x, t ) are derived. Four asymptotic regions are divided in this work: For ξ ∈ (cid:0) −∞ , − 14 (cid:1) ∪ (2 , ∞ ) , the phase function θ ( z ) has no stationary point on the jump contour, and the asymptotic approximations can be characterized with the soliton term conﬁrmed by N ( j 0 ) -soliton on discrete spectrum with residual error up to O ( t − 1+2 τ ) ; For ξ ∈ (cid:0) − 14 , 0 (cid:1) and ξ ∈ (0 , 2) , the phase function θ ( z ) has four and two stationary points on the jump contour, and the asymptotic approximations can be characterized with the soliton term conﬁrmed by N ( j 0 ) -soliton on discrete spectrum and the t − 12 order term on continuous spectrum with residual error up to O ( t − 1 ) . Our results also conﬁrm the soliton resolution conjecture for the CH equation with weighted Sobolev initial data in space-time solitonic regions.

21 Jun 2022

TL;DR: In this article , the authors employ the ¯ ∂ -steepest descent method to investigate the Cauchy problem of the nonlocal nonlinear Schrödinger (NNLS) equation with ﬁnite density type initial conditions in weighted Sobolev space H ( R ) .

Abstract: . In this work, we employ the ¯ ∂ -steepest descent method to investigate the Cauchy problem of the nonlocal nonlinear Schrödinger (NNLS) equation with ﬁnite density type initial conditions in weighted Sobolev space H ( R ) . Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the NNLS equation with the ﬁnite density type initial boundary value condition. By developing the ¯ ∂ -generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q ( x,t ) in soliton region of space-time, (cid:0) x 2 t (cid:1) = ξ for any ﬁxed ξ = ∈ (1 , K ) ( K is a suﬃciently large real constant), and give bounds for the error decaying as | t | → ∞ . Based on the resulting asymptotic behavior, the asymptotic approximation of the NNLS equation is characterized with the soliton term conﬁrmed by N (Λ) -soliton on discrete spectrum and the t − 12 order term on continuous spectrum with residual error up to O ( t − 34 ) .