Abstract
We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.
Highlights
The purpose of this short article is to present a classification of nonlinear partial differential equations of second order of the general form uxt = u + c0u2 + c1uux + c2uuxx + c3u2x + d0u3 + d1u2ux + d2u2uxx + d3uu2x, (1) which in the case that c j = 0 for j = 0, 1, 2, 3 and d0 = d1 = 0, d3 = 2d2, includes the short pulse equation derived by Schäfer and Wayne [29] as a model of ultra-short optical pulses in nonlinear media; cf
It was shown by Sakovich and Sakovich [27] that the short pulse equation is integrable, in the sense that it admits a Lax pair and a recursion operator that generates infinitely many commuting symmetries; these authors found a hodograph-type transformation connecting it with the sine-Gordon equation
The list of integrable generalized short pulse equations appears to contain three new equations, namely (4), (5), and (8), which combines the nonlinear terms of the Hunter–Saxton equation and the single-cycle pulse equation
Summary
The purpose of this short article is to present a classification of nonlinear partial differential equations of second order of the general form uxt = u + c0u2 + c1uux + c2uuxx + c3u2x + d0u3 + d1u2ux + d2u2uxx + d3uu2x , (1) which in the case that c j = 0 for j = 0, 1, 2, 3 and d0 = d1 = 0, d3 = 2d2, includes the short pulse equation derived by Schäfer and Wayne [29] as a model of ultra-short optical pulses in nonlinear media; cf Eq (3) below It was shown by Sakovich and Sakovich [27] that the short pulse equation is integrable, in the sense that it admits a Lax pair and a recursion operator that generates infinitely many commuting symmetries; these authors found a hodograph-type transformation connecting it with the sine-Gordon equation.
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