Reciprocal gausson phenomena in a Korteweg capillarity system

Abstract

In previous work in the literature, a kinetic derivation of a logarithmic nonlinear Schrödinger equation incorporating a de Broglie–Bohm term has been obtained in a capillarity context. Here, gausson-type solutions are constructed for such an NLS model. A novel two-parameter class of reciprocal transformations is shown to leave the 1 + 1-dimensional Korteweg capillarity system invariant and is applied to generate reciprocal gausson solutions. Extension of these results to q-gaussons is presented.

Appendix: The invariance property

$$\begin{aligned} \varDelta =\varDelta ^* \end{aligned}$$

Under the class of reciprocal transformations $R^*$ given by (27) wherein

$$\begin{aligned} \begin{array}{c} \rho ^*=\dfrac{\rho }{1+\chi \rho }\ , \quad \kappa ^*=(1+\chi \rho )^5\kappa (\rho ) \\ dx^*=(1+\chi \rho )dx-\chi \rho q\ dt\ , \quad t^*=t \end{array} \end{aligned}$$

it is seen that

$$\begin{aligned} \rho ^*_{x^*}= & {} \dfrac{1}{1+\chi \rho }\dfrac{\partial }{\partial x}\left[ \dfrac{\rho }{1+\chi \rho }\right] =\dfrac{\rho _x}{(1+\chi \rho )^3}, \\ \rho ^*_{x^*x^*}= & {} \dfrac{1}{1+\chi \rho }\dfrac{\partial }{\partial x}\left[ \dfrac{\rho _x}{(1+\chi \rho )^3}\right] =\dfrac{1}{1+\chi \rho }\left[ \dfrac{\rho _{xx}}{(1+\chi \rho )^3}-\dfrac{3\chi \rho ^2_x}{(1+\chi \rho )^4}\right] , \end{aligned}$$

together with

$$\begin{aligned} d\kappa ^*/d\rho ^*=(d\kappa ^*/d\rho )(d\rho /d\rho ^*)=[\ 5(1+\chi \rho )^4\chi \kappa (\rho )+(1+\chi \rho )^5\kappa '(\rho )\ ](1+\chi \rho )^2. \end{aligned}$$

Hence,

$$\begin{aligned} \begin{array}{l} \varDelta ^*=\kappa ^*(\rho ^*)\left( \rho ^*_{x^*x^*}-\dfrac{1}{2}\rho ^2_{x^*}\right) +\dfrac{1}{2}\rho ^*(d\kappa ^*/d\rho ^*)\rho ^2_{x^*}\\ \quad \quad =(1+\chi \rho )^5\kappa (\rho )\left[ \dfrac{1}{1+\chi \rho }\left( \dfrac{\rho _{xx}}{(1+\chi \rho )^3}-\dfrac{3\chi \rho ^2_x}{(1+\chi \rho )^4}\right) \dfrac{\rho }{1+\chi \rho }-\dfrac{1}{2}\ \dfrac{\rho ^2_x}{(1+\chi \rho )^6}\right] \\ \quad \quad \quad \quad +\dfrac{1}{2}\ \dfrac{\rho }{1+\chi \rho }[\ 5(1+\chi \rho )^6\kappa (\rho )+(1+\chi \rho )^7\kappa '(\rho )\ ]\dfrac{\rho ^2_x}{(1+\chi \rho )^6} \\ \quad \quad =\kappa (\rho )\left[ \rho _{xx}\rho +\dfrac{\rho ^2_x(-3\chi \rho -\tfrac{1}{2}+\tfrac{5}{2}\rho )}{1+\chi \rho }\right] +\dfrac{1}{2}\rho \kappa '(\rho )\rho ^2_x \\ \quad \quad =\kappa (\rho )\left( \rho _{xx}\rho -\dfrac{1}{2}\rho ^2_x\right) +\dfrac{1}{2}\rho \kappa '(\rho )\rho ^2_x=\varDelta \ . \end{array} \end{aligned}$$

Thus, the invariance property $\varDelta =\varDelta ^*$ is established.

It is noted, in conclusion, that aspects of another kind of reciprocal transformation allied with Bäcklund transformations have been investigated in connection with the Ermakov equation in [61]. The invariance of classical gasdynamic systems up to the equation of state under certain 4-parameter reciprocal transformations have been shown to constitute Bäcklund transformations in [28].