Related papers: Nonlinear stability of mKdV breathers
We derive a Hamiltonian version of the ${\cal PT}$-symmetric discrete nonlinear Schr\"{o}dinger equation that describes synchronized dynamics of coupled pendula driven by a periodic movement of their common strings. In the limit of weak…
We characterize a general traveling periodic wave of the defocusing mKdV (modified Korteweg--de Vries) equation by using a quotient of products of Jacobi's elliptic theta functions. Compared to the standing periodic wave of the defocusing…
Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be…
By means of a modified Lugiato-Lefever equation model, we investigate the nonlinear dynamics of dissipative wave structures in coherently-driven Kerr cavities with a parabolic potential. The potential stabilizes system dynamics, leading to…
We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vi\c{s}an-Zhang (2018) discovered a conserved quantity which they used to prove low…
We consider the vector generalization of the modified Korteweg-de Vries equation. We develop the inverse scattering transform for solving this equation. We construct the solitons and the breather solutions and investigate the processes of…
We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation $\partial_t u = -\partial_x (\partial_x^2 u + 3u^2-bu)$, where $b(x,t) = b_0(hx,ht)$, $h\ll 1$ is a slowly varying, but not small, potential. We option an…
In this paper we consider a discrete Klein-Gordon (dKG) equation on $\ZZ^d$ in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling…
Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov-Ma breathers and the Akhmediev breathers of the focusing nonlinear Schr{\"o}dinger equation. We use the Darboux transformation to…
The nonlinear Schr\"{o}dinger equation with variable coefficients has applications in numerous areas of physics,specifically in the context of nonlinear optics and Bose-Einstein condensate. Apart from the usual bright and dark-soliton…
We study metastable motions in weakly damped Hamiltonian systems. These are believed to inhibit the transport of energy through Hamiltonian, or nearly Hamiltonian, systems with many degrees of freedom. We investigate this question in a very…
We consider the discrete p-Schr\"odinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order alpha = p-1 >1. Using a mapping approach, we…
This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different…
Using two methods we show that a quantized discrete breather in a 1-D lattice is stable. One method uses path integrals and compares correlations for a (linear) local mode with those of the quantum breather. The other takes a local mode as…
We prove two new mixed sharp bilinear estimates of Schr\"odinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schr\"odinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}. Our…
We consider the Hamiltonian version of a $\cal PT$-symmetric lattice that describes dynamics of coupled pendula under a resonant periodic force. Using the asymptotic limit of a weak coupling between the pendula, we prove the nonlinear…
We consider the approximate control of solitons in generalized Korteweg-de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is locally null controllable, and moreover, any…
In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit $\lambda_{j}$ $\rightarrow$ $\lambda_{1}$ of the Lax pair…
The global stability of the nonhomogeneous positive steady state solution to a diffusive Holling-Tanner predator-prey model in a heterogeneous environment is proved by using a newly constructed Lyapunov function and estimates of nonconstant…
We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum P of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution p of (mKdV) such that p(t) -- P(t) $\rightarrow$ 0 when t…