Solutions of the Perturbed KDV Equation for Convecting Fluids by Factorizations

On integrability of a (2+1)-dimensional perturbed Kdv equation

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé test for integrability well, and its 4$\times $4 Lax pair with two spectral parameters is found. The results show that the Painlevé classification of coupled KdV equations by A. Karasu should be revised.

Khasminskii--Whitham averaging for randomly perturbed KdV equation

We consider the damped-driven KdV equation $$ \dot u-ν{u_{xx}}+u_{xxx}-6uu_x=\sqrtνη(t,x), x\in S^1, \int u dx\equiv \intηdx\equiv0, $$ where $0<ν\le1$ and the random process $η$ is smooth in $x$ and white in $t$. For any periodic function $u(x)$ let $ I=(I_1,I_2,...) $ be the vector, formed by the KdV integrals of motion, calculated for the potential $u(x)$. We prove that if $u(t,x)$ is a solution of the equation above, then for $0\le t\lesssimν^{-1}$ and $ν\to0$ the vector $ I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),...) $ satisfies the (Whitham) averaged equation.

On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation ter…

Generalized Solitary Waves in a Finite-Difference Korteweg-de Vries Equation

Generalized solitary waves with exponentially small non-decaying far field oscillations have been studied in a range of singularly-perturbed differential equations, including higher-order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behaviour of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behaviour in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher-order KdV equations a…

Dynamics of KdV solitons in the presence of a slowly varying potential

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 explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale $δh^{-1}\log h^{-1}$, together with an estimate on the error of size $h^{1/2}$. In addition to the Lyapunov analysis commonly applied to these problems, we use a local virial estimate due to Martel-Merle (2005). The results are support…

周期性海底地形对行进的Boussinesq孤子的扰动

Perturbation of Traveling Boussinesq Solitons by Periodic Bathymetry

研究周期性海底地形对Boussinesq孤子的扰动，通过两种方法分析扰动效应及稳定性。