On integrability of a third-order complex nonlinear wave equation

Abstract

We show that the new third-order complex nonlinear wave equation, introduced recently by Müller-Hoissen [arXiv:2202.04512], does not pass the Painlevé test for integrability. We find two reductions of this equation, one integrable and one non-integrable, whose solutions jointly cover all solutions of the original equation.

A vectorial binary Darboux transformation for the first member of the negative part of the AKNS hierarchy

“…In this sector, the only exact solution we know is the fairly trivial one f = Ce i α(x)−β t , where α is any real function with non-vanishing derivative, β = 0 is a real constant and C = 0 a complex constant. That switching on an imaginary part of the function a in (3.1) apparently breaks complete integrability, is an interesting observation [28], which deserves further exploration.…”

“…In this work we explored the nonlinear PDE (1.1), which is completely integrable (in the sense that a Lax pair exists) if the dependent variable is real. Expressing (1.1) as the equivalent system (3.1), in the complex case integrability apparently requires that the function a has to be real [28]. Accordingly, the vectorial binary Darboux transformation, which we generated from a universal binary Darboux transformation in bidifferential calculus, only works for (3.1) with the restriction to real a.…”

“…Although the first of equation (3.1) requires a t to be real, this does not exclude an imaginary part of a. As pointed out in [28], there is thus another reduction of (3.1), where a = a 1 + ia 2 , with real functions a j , j = 1, 2, a 2 = 0, a 2t = 0, and this does not pass the Painlevé test of integrability 4 .…”

“…If f (x, t) is allowed to be complex, then (1.1) is most likely not completely integrable [28]. But the reduction of (3.1), obtained by restricting the function a to be real, possesses a Lax pair (cf the remark 3.4 below) and is thus completely integrable in this sense.…”