Bäcklund connections and Bäcklund transformations corresponding to second order evolution equations (Q2276600)

The authors consider Bäcklund connections and Bäcklund transformations corresponding to the second-order evolution equations. A system of differential equations \(\Psi(x^i,z,y,z_k,y_l,\dots)=0\) is called a Bäcklund transformation from the differential equation \(F(x^i,z,z_j,z_{kl},\dots)=0\) to the differential equation \(\Phi(x^i,y,y_j,y_{kl},\dots)=0\), if for any solution \(z=z(x^1,\dots,x^n)\) of \(F=0\), \(\Psi=0\) determines a solution \(y=y(x^1,\cdots,x^n)\) of \(\Phi=0\). An invariant theory of Bäcklund transformations is first developed for the second-order evolution equations. The main result of the paper is about a special second-order evolution equation \(z_t-z_{xx}-\xi (z) (z_x)^2 -\eta (z) z_x=0\). It is proved that this differential equation admits a Bäcklund transformation of the type \[ \begin{aligned} y_t&=-yg^1_{01}(z,z_x)+\gamma ^1 _{00}(z,z_x),\\ y_x&=-y\gamma_{11}^1(z)+\gamma^1_{01}(z), \end{aligned} \] with \((\frac {d \gamma^1_{11}}{dz} )^2+(\frac {d\gamma^1_{01}}{dz})^2\neq 0\) if and only if it is either of the type \[ z_t-z_{xx}-\frac {\eta ''(z)}{\eta '(z)}(z_x)^2-\eta (z) z_x=0,\quad \eta '(z)\neq 0, \] or of the type \[ z_t-z_{xx}-\xi (z) (z_x)^2 -k z_x=0, \quad k=\text{const.} \] The equation of first type has the Bäcklund transformation \[ \begin{aligned} y_t&= (-y+c_1) \biggl[ \frac 12 \eta '(z)z_x+ \frac 14 \eta ^2(z) -\biggl( \frac {c_2}{2c}\biggr)^2 \biggr ] +c \biggl( \frac 12 \eta (z) +\frac {c_2}{ 2c}\biggr),\\ y_x&=(-y+c_1)\biggl(\frac 12 \eta (z)-\frac {c_2}{2c}\biggr)+c, \end{aligned} \] where \(c\neq 0,c_1\) and \(c_2\) are arbitrary constants, and becomes the Burgers equation \(Z_t-Z_{xx}+ZZ_x=0\) under the transformation of variables \(Z=-\eta (z)\). The equation of second type has the Bäcklund transformation \[ \begin{aligned} y_t&=(k-a)\biggl(-ay+c\int e^{F(z)}\, dz+c_1\biggr ) +c\, e^{F(z)}z_x,\\ y_x&=-ay +c \int e^{F(z)}\, dz +c_1, \end{aligned} \] where \(a\neq 0,c\neq 0,c_1\) are arbitrary constants and \(F'(z)=\xi (z)\), and becomes the linear equation \(Z_t-Z_{xx}-kZ_x=0\) under the transformation of variables \(Z=\int e^{F(z)}\, dz\). The paper covers topics of the current interest to integrable evolution equations, but only two examples of linearizable evolution equations which possess the required type of Bäcklund transformation are presented.