Related papers: The KdV hierarchy in optics
Periodic waves are investigated in a system composed of a Kuramoto-Sivashinsky - Korteweg-de Vries (KS-KdV) equation, which is linearly coupled to an extra linear dissipative equation. The model describes, e.g., a two-layer liquid film…
As a prototype of an evolution equation we consider the Schr\"odinger equation i (d/dt) \Psi(t) = H \Psi(t), H = H_0 + V(x) for the Hilbert space valued function \Psi(.) which describes the state of the system at time t in space dimension…
We study the conditions for the formation and propagation of Korteweg-de Vries (KdV) solitons in nuclear matter. In a previous work we have derived a KdV equation from Euler and continuity equations in non-relativistic hydrodynamics. In the…
Surface plasmon polaritons propagating along curved metal-dielectric interfaces experience geometry-induced modifications absent on flat surfaces. In this work, we derive a covariant, effective two-dimensional wave equation for the…
The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free…
We study the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ and $c_r$ as $x \to -\infty$ and $x\to+\infty$, respectively. Assuming $c_l>c_r>0$,…
Using the advective Cahn-Hilliard equation as a model, we illuminate the role of advection in phase-separating binary liquids. The advecting velocity is either prescribed, or is determined by an evolution equation that accounts for the…
Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials…
The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by…
The curvature of the inertial or gravitational potentials defined as a Hodge-Helmholtz decomposition of acceleration into an irrotational and a solenoidal components, enable to federate certain domains of macroscopic physics. After two…
The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general…
We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of…
We propose a novel on-surface radiation condition to approximate the outgoing solution to the Helmholtz equation in the exterior of several impenetrable convex obstacles. Based on a local approximation of the Dirichlet-to-Neumann operator…
Recently [A. Xuereb, et al., Phys. Rev. Lett. 105, 013602 (2010)], we calculated the radiation field and the optical forces acting on a moving object inside a general one-dimensional configuration of immobile optical elements. In this…
We study the integrability of mappings obtained as reductions of the discrete Korteweg-de Vries (KdV) equation and of two copies of the discrete potential Korteweg-de Vries equation (pKdV). We show that the mappings corresponding to the…
We propose a new method for calculating reflection and transmission coefficients for an arbitrarily polarized electromagnetic plane wave incident on a one-dimensional dielectric medium of finite thickness and with dielectric permittivity…
Under certain mode-matching conditions, small-amplitude waves can be trapped by coupling to solitons of nonlinear fields. We present a model for this phenomenon, consisting of a linear equation coupled to the Korteweg-de Vries equation. The…
We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for…
We announce a detailed investigation of limits of N-soliton solutions of the Korteweg-deVries (KdV) equation as $N$ tends to infinity. Our main results provide new classes of KdV-solutions including in particular new types of soliton-like…
A novel geometric method is applied to the problem of describing traveling wave solutions of the generalized Korteweg--de Vries (gKdV) equation in the form $$ u_t + u_{xxx} + a(u)u_x = 0, $$ where $a(u)$ is a smooth function characterizing…