In this paper, an all-optical plasmonic multi-wavelength switch based on Kerr nonlinear material is proposed. It consists of circular waveguides wrapped around three side-coupled nano-ring resonators. Fundamentally, introducing the circular waveguide increases the coupling coefficient and switching modulation depth. The transmission response of the proposed multi-switching structure is studied theoretically based on coupled mode and transfer matrix theories. The validity of the derived transmission formula is confirmed by the numerical result obtained by the finite element method. Also, based on the self-phase modulation and cross-phase modulation (XPM) nonlinear effects, the resonance wavelengths are effortlessly tuned by changing the intensity of the incident lightwave without changing the dimensions of the structure. As a result, by utilizing the XPM effect, the required input signal intensity is significantly decreased to 6.5MW/cm2. The obtained modulation depths are 18.08, 31.83, and 28.40 dB at wavelengths of 850, 1310, and 1550 nm, respectively. Finally, to show the application of the proposed switch, the simultaneous AND and NOR logic gates are designed with intensity contrast ratios of 78.81 and 85.49 dB, respectively. The proposed plasmonic switch has many advantages such as being multi-wavelength and having low required switching intensity, ultra-fast switching time of 23 fs, and optical bistability. These features are promising for future integrated plasmonic devices for applications such as communications, signal processing, and sensing.