We consider the Hamiltonian of α, β-Fermi Pasta Ulam lattice and explore the Hamilton–Jacobi formalism to obtain the discrete equation of motion. By using the continuum limit approximations and incorporating some normalized parameters, the extended Korteweg–de Vries equation is obtained, with solutions that elucidate on the Fermi Pasta Ulam paradox. We further derive the nonlinear Schrdinger amplitude equation from the extended Korteweg–de Vries equation, by exploring the reductive perturbative technique. The dispersion and nonlinear coefficients of this amplitude equation are functions of the α and β parameters, with the β parameter playing a crucial role in the modulational instability analysis of the system. For β greater than or equal to zero, no modulational instability is observed and only dark solitons are identified in the lattice. However for β less than zero, bright solitons are traced in the lattice for some large values of the wavenumber. Results of numerical simulations of both the Korteweg–de Vries and nonlinear Schrdinger amplitude equations with periodic boundary conditions clearly show that the bright solitons conserve their amplitude and shape after collisions.

我们考虑α , β -Fermi Pasta Ulam 晶格的哈密顿量，并探索 Hamilton-Jacobi 形式以获得离散的运动方程。通过使用连续极限近似并结合一些归一化参数，得到了扩展的 Korteweg-de Vries 方程，其解阐明了 Fermi Pasta Ulam 悖论。我们通过探索还原微扰技术，从扩展的 Korteweg-de Vries 方程进一步推导出非线性薛定谔振幅方程。该幅度方程的色散和非线性系数是α和β参数的函数，其中β参数在系统的调制不稳定性分析中起着至关重要的作用。对于大于或等于零的β ，没有观察到调制不稳定性，并且在晶格中仅识别出暗孤子。然而，对于小于零的β ，对于一些较大的波数值，在晶格中追踪明亮的孤子。Korteweg-de Vries 和具有周期性边界条件的非线性薛定谔振幅方程的数值模拟结果清楚地表明，明亮的孤子在碰撞后保持其振幅和形状。