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# Long time asymptotics for the defocusing mKdV equation with finite density initial data in different solitonic regions

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.

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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 ) .

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TL;DR: In this article, the authors present an approach to analyze the asymptotics of oscillatory Riemann-Hilbert problems with respect to the modified Korteweg-de Vries (MKdV) equation.

Abstract: In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation

1,409 citations

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Clarkson College

^{1}TL;DR: In this article, the inverse scattering method was used to solve the initial value problem for a broad class of nonlinear evolution equations, including sine-Gordon, sinh-Gordon and Benney-Newell.

Abstract: We present the inverse scattering method which provides a means of solution of the initial-value problem for a broad class of nonlinear evolution equations. Special cases include the sine-Gordon equation, the sinh-Gordon equation, the Benney-Newell equation, the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, and generalizations.

925 citations

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TL;DR: In this paper, the authors present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems, in particular in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method.

Abstract: In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation,
$$y_t-6y^2y_x+y_{xxx}=0,\qquad -\infty

728 citations

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Yale University

^{1}TL;DR: In this article, the authors present resultats sur la theorie analytique des problemes de diffusion and de diffusion inverse for des systemes generalises AKNS. But they do not consider diffusion in general.

Abstract: On considere le probleme aux valeurs propres suivant: df/dx=2Jf(x)+q(x)f(x). Avec f: R→C n , J est une matrice constante (n×n) et q est une fonction a valeur matricielle. On presente des resultats sur la theorie analytique des problemes de diffusion et de diffusion inverse pour des systemes generalises AKNS

566 citations

### "Long time asymptotics for the defoc..." refers methods in this paper

...The analysis of the RH problem is based on the so-called Beals-Coifman operator theory [30]....

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TL;DR: In this article, the system of equations for the magneto-acoustic wave propagating along a critical direction is reduced to a simple dispersive equation similar to the Korteweg-de Vries equation except that the third order derivative (the dispersion term) is replaced by the fifth order one.

Abstract: The hydromagnetic waves with small but finite amplitude in a cold collision-free plasma are investigated by using a nonlinear perturbation method. In the lowest order of perturbation, we can show that the system of equations for the magneto-acoustic wave propagating along a `critical' direction is reduced to a simple dispersive equation similar to the Korteweg-de Vries equation except that the third order derivative (the dispersion term) is replaced by the fifth order one. An extension of the problem to more general dispersive system is also made. On the other hand, the system of equations for the Alfven wave is reduced to a modified Korteweg-de Vries equation in the sense that the non-linear term f ∂ f /∂ξ in the Korteweg-de Vries equation is replaced by f 2 ∂/∂ξ. In the case of steady propagation this equation can be integrated to give a solution in closed form, which exhibits a solitary wave. Two kinds of solitary wave (both compressive and rarefied) are found to be possible.

363 citations

### "Long time asymptotics for the defoc..." refers background in this paper

...The mKdV equation arises in various of physical fields, such as acoustic wave and phonons in a certain anharmonic lattice [1, 2], Alfvén wave in a cold collision-free plasma [3, 4]....

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