arXiv subjects

Jiansheng Geng

Publications and source records attributed to Jiansheng Geng.

4 recordsLinked to original sources

arXiv2025

Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions

We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with periodic boundary conditions \[ u_{tt}-(1+\int_{0}^{2π} |u_x|^2 dx)u_{xx}+ M_ξu+εg(\barωt,x) =0,\quad u(t,x+2π)=u(t,x),\] where $M_ξ$ is a real Fourier multiplier, $g(\barωt,x)$ is real analytic with forced Diophantine frequencies $\barω$, $ε$ is a small parameter. The paper generalizes the previous results from the simple eigenvalue to the double eigenvalues under the quasi-linear perturbation.

math.DS↗

arXiv2019

A KAM Theorem for Two-dimensional Nonlinear Schrödinger Equations

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schrödinger equation $$iu_t-\triangle u +|u|^2u+\frac{\partial{f(x,u,\bar u)}}{\partial{\bar u}}=0, \quad t\in\Bbb R, x\in\Bbb T^2$$ with periodic boundary conditions, where the nonlinearity $\displaystyle f(x,u,\bar u)=\sum_{j,l,j+l\geq6}a_{jl}(x)u^j\bar u^l$, $a_{jl}=a_{lj}$ is a real analytic function in a neighborhood of the origin. We obtain for the equation a Whitney smooth family of small--amplitude quasi--periodic solutions which are partially hyperbolic.

math.DS↗

arXiv2019

A KAM Theorem for Higher Dimensional Reversible Nonlinear Schrödinger Equations

In the paper, we prove an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems. Using this KAM theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of reversible (non-Hamiltonian) coupled nonlinear Schrödinger systems on $d-$torus $\mathbb{T}^d$.

math.DS↗

arXiv2017

An infinite dimensional KAM theorem with application to two dimensional completely resonant beam equation

In this paper we consider the completely resonant beam equation on \T^2 with cubic nonlinearity on a subspace of L^2 (\T^2) which will be explained later. We establish an abstract infinite dimensional KAM theorem and apply it to the completely resonant beam equation. We prove the existence of a class of Whitney smooth small amplitude quasi-periodic solutions corresponding to finite dimensional tori.

math.DS↗