A second-order Korteweg–de Vries equation for a lattice

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Abstract
An analysis is made of the longitudinal response of a semi‐infinite lattice with a velocity disturbance at its end. Weakly nonlinear waves are analyzed, with a large acceleration time specified at an end to achieve a maximum disturbance. Nonlinear force interactions of quadratic and cubic form are considered within the lattice. A second‐order generalization of the Korteweg–de Vries equation is obtained for the far‐field response. The equation contains two stretched coordinates; one for the lowest‐order influence of quadratic nonlinearity, and the second for a combination of cubic and the second‐order influence of quadratic nonlinearity. Numerical solutions for shock profiles are given to illustrate the first‐ and second‐order effects of quadratic nonlinearity. The combined response with cubic nonlinearity is then illustrated.