LAUSR

In this paper, we consider the perturbed KdV equation with Fourier multiplier $$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$ut=-uxxx+(Mξu+u3)x,u(t,x+2π)=u(t,x),∫02πu(t,x)dx=0,with analytic data of size $$\varepsilon $$ε. We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with $$\tilde{J}$$J~ Diophantine frequencies, where the order of $$\tilde{J}$$J~ is $$O(\frac{1}{\varepsilon })$$O(1ε). The proof is based on a conserved quantity $$\int _0^{2\pi } u^2 dx$$∫02πu2dx, Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity $$\int _0^{2\pi } u^2 dx$$∫02πu2dx and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.