Lie symmetries, group classification and conserved quantities of dispersionless Manakov–Santini system in (2+1)-dimension

Abstract

A member of Manakov–Santini (MS) hierarchy is investigated in this work using Lie group analysis and the multiplier approach. The admitted 11-dimensional Lie algebra for the MS system has been proved to be completely solvable on basis of the existence of chain of ideals. The optimal list of inequivalent one-dimensional subalgebras are constructed from adjoint actions collected in a table. The method for construction of similar list in 2-dimension has also been discussed in detail. The subalgebras so obtained are used to give out several inequivalent reductions and subsequently some exact solutions are reported. In addition to usual Lie symmetry analysis, the infinite set of non-trivial conservation laws are obtained.

Appendix

$$\begin{aligned} \chi _{1}(x,y)=&\frac{ \left( a_{{3}}a_{{10}}-{a_{{10}}}^{2} \right) x-ya_{{9}}a_{{10}}-a_{{9}}}{a_{{10}} \left( a_{{3}}-a_{{10}} \right) } \end{aligned}$$

(43)

$$\begin{aligned} \chi _{{2}} \left( x,y \right) =&\frac{1}{ \left( a_{{3}}-2\,a_{{10}} \right) \left( a_{{3}}- a_{{10}} \right) ^{2} \left( 2\,a_{{3}}-a_{{10}} \right) }\Big (a_{{9}} \left( \left( 4\,{a_{{ 3}}}^{3}-14\,{a_{{3}}}^{2}a_{{10}}+14\,a_{{3}}{a_{{10}}}^{2}-4\,{a_{{ 10}}}^{3} \right) \right. x \nonumber \\&\left. + \left( -2\,{a_{{3}}}^{2}a_{{9}}+5\,a_{{3}}a_{{9}} a_{{10}}-2\,a_{{9}}{a_{{10}}}^{2} \right) y+2\,a_{{3}}a_{{9}}-a_{{9}}a _{{10}} \right) \Big ) \end{aligned}$$

(44)

$$\begin{aligned} \chi _{{3}} \left( x,y,t \right) =&-\ln \left( t \right) \left( \ln \left( t \right) a_{{8}}a_{{9}}t+\ln \left( t \right) {a_{{9}}}^{2}y -a_{{4}}t-2\,xa_{{9}}-a_{{5}}y \right) \end{aligned}$$

(45)

$$\begin{aligned} \chi _{{4}} \left( x,y,t \right) =&\,{\frac{t \left( -8\,{t}^{2}a_{ {5}}a_{{6}}+24\,ya_{{5}}{a_{{6}}}^{2}+9\,{t}^{3}-36\,tya_{{6}}+72\,x{a _{{6}}}^{2} \right) }{24{a_{{6}}}^{3}}}\end{aligned}$$

(46)

$$\begin{aligned} \chi _{{5}} \left( x,y,t \right) =&-\,{\frac{t \left( 4\,{t}^{2}a_{{ 8}}-3\,ta_{{4}}a_{{6}}-6\,ya_{{5}}a_{{6}}+6\,ty-12\,xa_{{6}} \right) }{6{a_{{6}}}^{2}}}\end{aligned}$$

(47)

$$\begin{aligned} \chi _{{6}} \left( x,y,t \right) =&\,{\frac{t \left( -8\,{t}^{2}a_{ {5}}a_{{6}}+24\,ya_{{5}}{a_{{6}}}^{2}+9\,{t}^{3}-36\,tya_{{6}}-36\,ta_ {{7}}a_{{6}}+72\,x{a_{{6}}}^{2} \right) }{24{a_{{6}}}^{3}}}\end{aligned}$$

(48)

$$\begin{aligned} \chi _{{7}} \left( x,y,t \right) =&-{\frac{t \left( -ya_{{5}}a_{{6}}+ty +a_{{7}}t-2\,xa_{{6}} \right) }{{a_{{6}}}^{2}}}\end{aligned}$$

(49)

$$\begin{aligned} \chi _{{8}} \left( x,y,t \right) =&-\,{\frac{t \left( {t}^{2}a_{{5}} -3\,a_{{5}}ya_{{6}}-3\,xa_{{6}} \right) }{3{a_{{6}}}^{2}}}\end{aligned}$$

(50)

$$\begin{aligned} \chi _{{9}} \left( x,y,t \right) =&-\,{\frac{x \left( xa_{{5}}t-2\,a _{{5}}ya_{{7}}-xa_{{7}} \right) }{2{a_{{7}}}^{2}}}\end{aligned}$$

(51)

$$\begin{aligned} \alpha =&\,\frac{2a_{9}^2}{(a_{3}-a_{10})(a_{3}-2a_{10})},\,\,\,\beta =\,\frac{a_{9}}{a_{3}-a_{10}},\,\,\, \gamma =\,\frac{a_{3}a_{9}}{a_{3}-a_{10}},\,\,\,\delta =\,\frac{2a_{9}^{2}}{a_{10}(a_{3}-a_{10})(a_{3}-2a_{10})}\end{aligned}$$

(52)

$$\begin{aligned} \theta =&\,\frac{a_{9}}{a_{10}(a_{3}-a_{10})} \end{aligned}$$

(53)