The Gaussian soliton in the Fermi-Pasta-Ulam chain

We consider a Fermi-Pasta-Ulam (FPU) chain with the homogeneous fully nonlinear interaction potential which describes the propagation of acoustic wave in chains of touching beads without precompression. From the quasi-continuum approximation of the FPU chain, we first derive out a new type of wave equation which includes a second degree logarithmic nonlinear term. By finding an integrable factor equation, we obtain its Gaussian solitary wave solution. The result shows that if the effect of logarithmic nonlinearity can be balanced with the dispersion, the Gaussian solitary waves do exist for the second degree logarithmic wave equation in real physical models.

Automated summary

We investigated the propagation of acoustic waves in chains of touching beads without precompression using a Fermi-Pasta-Ulam (FPU) chain with a homogeneous fully nonlinear interaction potential. By deriving a new wave equation with a second degree logarithmic nonlinear term and finding its Gaussian solitary wave solution through an integrable factor equation, we showed the existence of Gaussian solitary waves for the second degree logarithmic wave equation in real physical models when the effect of logarithmic nonlinearity is balanced with dispersion.