# Download PDF by Routh E.J.: A treatise on the dynamics of a system of rigid bodies

By Routh E.J.

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7. The numerical simulation of the interaction of two solitons using t t+1 t − u tn = δ(u t+1 the fully discrete soliton equation, u t+1 n n−1 u n − u n u n+1 ), where n = . . , −1, 0, 1, . . and δ is the lattice spacing. At t = 0, a large soliton is about to interact with a small soliton. At t = 28 they are overlapping, and at t = 49 they recover their original shapes. Their shapes are not smooth because their time evolutions take place at discrete time and space steps. where F is a general polynomial in Dt , Dx , D y , .

Predators travelling at speed c2 (> c1 ) catch up with prey travelling at speed c1 and their population size increases. Thin and thick lines are used for the size of predators and prey populations, respectively. At t = 0, the leading predator comes into contact with the ﬁrst of the prey. At t = 4, 5, 6, the predators eat the prey and increase in number. At t = 12, the two groups separate. Afterwards, the size of each group is unchanged. 4, we showed that certain nonlinear differential equations can be transformed into linear differential equations through a change of dependent variable.

202) Dx2 a · b = λab. The decoupling parameter λ can be chosen to be zero when seeking solitary wave solutions. 204) where f ∗ is the complex conjugate of f . 204) may be rewritten as sin φ = 1 2i f2 f ∗2 − f2 f ∗2 . 204) gives [i(Dx2 − Dt2 ) f · f − ( f ∗2 − f 2 )/(2i)] f ∗2 − [i(Dx2 − Dt2 ) f ∗ · f ∗ − ( f 2 − f ∗2 )/(2i)] f 2 = 0. 207) together with its complex conjugate. 208) the sine–Gordon equation, φx x − φtt = sin φ, is transformed into φx y = sin φ. 211) as where λ is real and is chosen to be zero when seeking solitary wave solutions.