Related papers: Long time decay and asymptotics for the complex mK…
Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution…
We consider perturbations of the special pole-free joint solution $U(x,t)$ of the Korteweg--de Vries equation $u_t+uu_x+\frac{1}{12}u_{xxx}=0$ and $P_I^2$ equation $u_{xxxx}+10u_x^2+20uu_{xx}+40(u^3-6tu+6x)=0$ under the action of the KdV…
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{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0,…
We investigate the initial value problem (IVP) associated to the modified Korteweg-de Vries equation (mKdV) in the defocusing scenario: \begin{equation*} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u-u^2\partial_x(u) = 0, \quad…
This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on…
We review the theory of modulation equations or Whitham equations for the travelling wave solution of KdV. We then apply the Whitham modulation equations to describe the long-time asymptotics and small dispersion asymptotics of the KdV…
In this paper, we investigate the quantitative exponential stability of the Korteweg-de Vries equation on a finite interval with its length close to the critical set. Sharp decay estimates are obtained via a constructive PDE control…
Using Levi-Civita's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity…
It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry…
In this work, we use scattering method to study the Kramers-Fokker-Planck equation with a potential whose gradient tends to zero at the infinity. For short-range potentials in dimension three, we show that complex eigenvalues do not…
We have constructed one-phase quasi-periodic solutions of the curve equation induced by the mKdV equation. The solution is expressed in terms of the elliptic functions of Weierstrass. This solution can describe curve dynamics such as a…
We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this…
Both Sawada-Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are integrable systems with third-order Lax operator. Moreover, they are related with the same modified nonlinear equation (called modified SK-KK equation) by Miura…
We utilize a modulation restricted normal form approach to establish local well-posedness of the periodic Korteweg-de Vries equation in $H^s(\mathbb{T})$ for $s> -\frac23$. This work creates an analogue of the mKdV result by Nakanishi,…
We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is…
We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +\lambda_0(x) u(x,t)+\lambda(x) u(x,t-\tau )=0 \end{equation*} where…
Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other…
In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-…
We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schr\"odinger equation by means of the Deift-Zhou nonlinear steepest descent method. The leading term is a sum of two terms that oscillate with decay…
Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg-de Vries equation posed on a finite interval $\left[0, 2\pi \sqrt{7/3}\right]$. The equation comes with a Dirichlet boundary condition at…