The direct method in soliton theory

By 広田良吾, 永井敦, J. Nimmo, C. Gilson. 2004

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Expanding the Tanh-Function Method for Solving Nonlinear Equations (2011, methodology): “Many powerful methods have been developed to find the exact solutions of nonlinear PDEs, among them inverse scattering method [1], Hirota bilinear form [2], Painlevé analysis [3], direct algebraic method [4], tanh-function method [5,6] and it’s extensions [7-9] and the sine-cosine method [10,11].”
On the Camassa–Holm equation and a direct method of solution. III. N-soliton solutions (2005, methodology): “A (2005) Our direct method makes use of Hirota’s bilinear transformation theory (Hirota 1980) which remains a powerful tool for finding exact solutions of nonlinear evolution equations (Hirota 2004).”
An integrable symmetric (2+1)-dimensional Lotka–Volterra equation and a family of its solutions (2005, methodology): “where the bilinear operators D t and exp(Dn) are defined by [17, 18, 19] Dt a • b ≡ ( ∂ ∂t − ∂ ∂t′ ) a(t)b(t′)|t′=t, exp(δDn)a(n) • b(n) = a(n + δ)b(n− δ).”
Abundant Mixed Lump-Soliton Solutions to the BKP Equation (2018, methodology): “It is worth noting that the general rational solutions of integrable equations have been derived within the framework of Wronskian, Casoratian, Grammian and Pfaffian formulations [1, 16].”