Lax pair of an integrable non-linear PDE

Question

The following is a fourth-order non-linear PDE that passes the Painleve integrability test $$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(2xuu_{xx} + \left(1+x^{2}\right)\left(uu_{xx}\right)_{x} - 4\left(1+3x^{2}\right)u_{xxt} - 4x\left(1+x^{2}\right)u_{xxxt}\right)=0,$$

where $u=u(x,t)$. The leading-order behaviour of this PDE is of the form $u=A_{0}\left(x-x_{0}\right)^{-1}$, where $A_{0}=3\left(1+x_{0}^{2}\right)t^{-1}$. I am looking for a Lax pair $[\mathcal{L},\mathcal{M}]$ which satisfies the equation $$\dot{\mathcal{L}}+[\mathcal{L},\mathcal{M}]=0,$$

but not able to find the same. I suspect that this may be due to the fact that although few systems may pass the Painleve test, they need not be integrable. Any help/comment(s) about the problem and/or the integrability of the PDE is appreciated. Also, are there any other (stronger) methods using which I can explicitly try and probe for the integrability of the PDE although it passed the Painleve test?

Answer 1

You could try using the Wahlquist-Estabrook prolongation structure technique, per H.D. Wahlquist and F.B. Estabrook, J. Math. Phys 16 (1975) 1-7 (covering the Korteweg-deVries equation), & F.B. Estabrook and H.D. Wahlquist J. Math. Phys. 17 (1976) 1293-1297 (covering the nonlinear Schrödinger equation).

A non-trivial prolongation structure would be a signal that your equation is indeed integrable. To construct a Lax pair, you need to find an explicit representation of the prolongation structure: the paper by R. Dodd and A. Fordy Proc Roy. Soc. Lond. A385 (1983) 389-429 provides a method of doing this.

Since your PDE includes both independent variables in the coefficients, you may also wish to review the note by B.A. Kupershmidt, Nonautonomous form of the theory of Lax equations, Lettere al Nuovo Cimento 33 (1982) 103–107.