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Chapter 21: Hirota Technique and Painlevé Test

https://doi.org/10.1142/9789813275386_0021Cited by:0 (Source: Crossref)

Abstract:

The Hirota technique plays a central role in finding solutions to soliton equations. Let f, g be smooth functions. The Hirota bilinear operators D_x and D_t are defined as

$D_{t}^{n} D_{x}^{m} (f \circ g) : = {(\frac{\partial}{\partial t} - \frac{\partial}{\partial t^{'}})}^{n} {(\frac{\partial}{\partial x} - \frac{\partial}{\partial x^{'}})}^{m} f (t, x) g (t^{'}, x^{'}) |_{x^{'} = x, t^{'} = t}$

where m, n = 0, 1, 2, …. The Hirota bilinear operator is linear. From this definition it follows that

$\begin{array}{c} D_{x}^{m} (f \circ f) = 0 for odd m \\ D_{x}^{m} (f \circ g) = {(- 1)}^{m} D_{x}^{m} g \circ f \\ D_{x} (f \circ g) = \frac{\partial f}{\partial x} g - f \frac{\partial g}{\partial x} \\ D_{t} (f \circ g) = \frac{\partial f}{\partial t} g - f \frac{\partial g}{\partial t} \\ \frac{\partial^{2}}{\partial x^{2}} I n (f) = \frac{1}{2 f^{2}} (D_{x}^{2} f \circ f) \\ \frac{\partial^{2}}{\partial x \partial t} I n (f) = \frac{1}{2 f^{2}} D_{x} D_{t} f \circ f \\ \frac{\partial^{4}}{\partial x^{4}} I n (f) = \frac{1}{2 f^{2}} D_{x}^{4} (f \circ f) - 6 {(\frac{1}{2 f^{2}} D_{x}^{2} (f \circ f))}^{2} . \end{array}$