A Higher-Order Korteweg-de Vries Equation

The higher-order KdV equation [8]

$$u_t={1\over 4}{\partial\over\partial_x}(u_{xxxx}+5u_x^2+10u_{xx}u+ 10u^3)\eqno (5.1)$$

has an expansion, about the singular manifold, of the form

$$u=\phi^{-2}\sum_{j=0}^{\infty} u_j\phi^j .\eqno (5.2)$$

The resonances occur at

(i)$r=-1,2,5,6,8,$when $u_0=-2 \phi_x^2$(5.3)

(ii) $r=-3,-1,6,8,10,$    when $u_0=-6\phi_x^2 .$(5.4)

We note the appearance of two solution branches, with branch (i) depending on five, and branch (ii) depending on four, arbitrary functions [1], [3]. In this section we only consider branch (i).

Rather than verify by tedious calculation that the equation possesses the Painlevé property, we shall assume that the equation has an associated Bäcklund transform

$$u=-2\phi_x^2/\phi^2+u_1/\phi+u_2 .\eqno (5.5)$$

By substitution in (5.1) we find that

$$u_0=-2\phi_x^2 ,\qquad u_1=2\phi_{xx} ,\eqno (5.6)$$

$$u_2=-{1\over 2}\left[{\partial\over\partial x}\left({\phi_{xx... ...({\phi_{xx}\over\phi_x}\right)^2\right] +\lambda ,\eqno (5.7)$$

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...\phi_{xx}^2\over\phi_x^2}\right) u_2 ,\cr\crcr}} \eqno (5.8)$$

and $u_2$ must satisfy (5.1).

To verify the consistency of the above Bäcklund transform, we substitute (5.7) and (5.8) and obtain

$$4{\phi_t\over\phi_x}={\partial^2\over\partial x^2}\{\phi;x\}+... ...er 2}\{\phi;x\}^2+10\lambda\{\phi;x\}+30\lambda^2 ,\eqno (5.9)$$

where

$$\{\phi;x\}={\partial\over\partial x}\left({\phi_{xx}\over\phi_x} \right)-{1\over 2}\left({\phi_{xx}\over\phi_x}\right)^2$$

is the Schwarzian derivative.

From Eq. (5.9) we readily obtain the Lax pair. Letting

$$\phi=v_1/v_2 ,\eqno (5.10)$$

and

$$v_{xx}=av ,\qquad v_t=bv_x+cv ,\eqno (5.11)$$

we find from (5.9) that

$$4b=-2a_{xx}+6a^2-20\lambda a+30\lambda^2 .\eqno (5.12)$$

With the consistency conditions

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ..._{xx}+2ab_x+ba_x ,\cr &a=\lambda-u ,\cr\crcr}} \eqno (5.13)$$

we find

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... c&=-\textstyle{1\over 2}b_x+\alpha .\cr\crcr}} \eqno (5.14)$$

This defines the Lax pair for (5.1) [8].