Given PDESYS, a system of equations for DepVars, possibly including inequations and non-differential equations, PolynomialSolutions computes solutions that are polynomial in the independent variables of the problem. The independent variables are typically the variables DepVars depend on, plus any others optionally specified with ivars = .... PolynomialSolutions also works with anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.

If DepVars is not given, PolynomialSolutions will consider all the differentiated unknown functions in PDESYS as unknown of the problems. Specifying DepVars however permits not only restricting the unknowns in different ways but also specifying unknowns of the problems which do not appear differentiated in PDESYS.

PolynomialSolutions uses a heuristic algorithm to compute upper bound degrees. The algorithm used extends, for non-autonomous systems of equations, the ideas used in TWSolutions (step 2. of The solving process) for computing solutions to autonomous systems. This extended algorithm is accessible at user level as PDEtools:-Library:-UpperBounds (see PDEtools,Library). To override these upper bounds see the degree option.

PolynomialSolutions is automatically invoked by Infinitesimals, InvariantSolutions and SimilaritySolutions when they receive the optional argument typeofsymmetry = polynomial in which case the dependency and degree can also be specified directly to those commands.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

with(PDEtools):

declare((xi, eta)(x, y));

$\mathrm{\xi }\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\xi }$

$\mathrm{\eta }\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\eta }$

sys[1] := diff(xi(x,y),x)*diff(eta(x,y),x)+diff(xi(x,y),y)*diff(eta(x,y),y)+eta(x,y)*(diff(xi(x,y),x,x)+diff(xi(x,y),y,y)) = -1;

${\mathrm{sys}}_1≔{\mathrm{\xi }}_x{\mathrm{\eta }}_x+{\mathrm{\xi }}_y{\mathrm{\eta }}_y+\mathrm{\eta }\left({\mathrm{\xi }}_{x,x}+{\mathrm{\xi }}_{y,y}\right)=\mathrm{-1}$

sol[1] := PolynomialSolutions(sys[1]);

${\mathrm{sol}}_1≔\left\{\mathrm{\eta }=\frac{\left(\mathrm{c\_\_3}{y}^2+\mathrm{c\_\_2}y+\mathrm{c\_\_1}\right)\mathrm{c\_\_5}-x}{\mathrm{c\_\_5}}\,\mathrm{\xi }=\mathrm{c\_\_5}x+\mathrm{c\_\_4}\right\},\left\{\mathrm{\eta }=\frac{\left(\mathrm{c\_\_3}{x}^2+\mathrm{c\_\_2}x+\mathrm{c\_\_1}\right){\mathrm{c\_\_5}}^2+\left(-2\mathrm{c\_\_3}\mathrm{c\_\_6}x-\mathrm{c\_\_2}\mathrm{c\_\_6}-1\right)y\mathrm{c\_\_5}+\mathrm{c\_\_3}{\mathrm{c\_\_6}}^2{y}^2}{{\mathrm{c\_\_5}}^2}\,\mathrm{\xi }=\mathrm{c\_\_5}y+\mathrm{c\_\_6}x+\mathrm{c\_\_4}\right\}$

map(pdetest, [sol[1]], sys[1]);

$\left[0\,0\right]$

PolynomialSolutions(sys[1], degree = 1, dependency = 1);

PolynomialSolutions(sys[1], dependency = 0..2);

InvariantSolutions(sys[1], typeofsymmetry = polynomial, degree = 0);

$\left\{\mathrm{\eta }=\frac{-x+\mathrm{c\_\_1}}{{\mathrm{f\_\_1}}_x}\,\mathrm{\xi }=\mathrm{f\_\_1}\left(x\right)\right\},\left\{\mathrm{\eta }=\frac{-y+\mathrm{c\_\_1}}{{\mathrm{f\_\_1}}_y}\,\mathrm{\xi }=\mathrm{f\_\_1}\left(y\right)\right\}$

sys[2] := [-I*x^3*(diff(xi(x,y),y,y)*y+3*diff(xi(x,y),y))/y, -I*x^2*(-diff(eta(x,y),y,y)*x*y^2+2*diff(xi(x,y),x,y)*x*y^2-3*eta(x,y)*x+3*x*diff(eta(x,y),y)*y+2*diff(xi(x,y),y)*y^2)/y^2, x*(2*I*diff(eta(x,y),x,y)*x^2*y-I*diff(xi(x,y),x,x)*x^2*y-I*x*diff(xi(x,y),x)*y-6*I*x^2*diff(eta(x,y),x)+6*diff(xi(x,y),y)*y^4*ln(x)+3*diff(xi(x,y),y)*y^2+xi(x,y)*y*I)/y, -4*xi(x,y)*y^3*ln(x)-2*xi(x,y)*y+6*eta(x,y)*x*ln(x)*y^2-I*x^2*diff(eta(x,y),x)+4*x*diff(xi(x,y),x)*y^3*ln(x)+2*x*diff(xi(x,y),x)*y-2*x*diff(eta(x,y),y)*y^3*ln(x)-x*diff(eta(x,y),y)*y+2*xi(x,y)*y^3+diff(eta(x,y),x,x)*x^3*I+eta(x,y)*x];

${\mathrm{sys}}_2≔\left[\frac{\mathrm{-I}{x}^3\left({\mathrm{\xi }}_{y,y}y+3{\mathrm{\xi }}_y\right)}{y}\,\frac{\mathrm{-I}{x}^2\left(-{\mathrm{\eta }}_{y,y}x{y}^2+2{\mathrm{\xi }}_{x,y}x{y}^2-3\mathrm{\eta }x+3x{\mathrm{\eta }}_{y}y+2{\mathrm{\xi }}_y{y}^2\right)}{y^2}\,\frac{x\left(2\mathrm{I}{\mathrm{\eta }}_{x,y}{x}^{2}y-\mathrm{I}{\mathrm{\xi }}_{x,x}{x}^{2}y-\mathrm{I}x{\mathrm{\xi }}_{x}y-6\mathrm{I}{x}^2{\mathrm{\eta }}_x+6{\mathrm{\xi }}_y{y}^4\ln \left(x\right)+3{\mathrm{\xi }}_y{y}^2+\mathrm{I}\mathrm{\xi }y\right)}{y}\,-4\mathrm{\xi }{y}^3\ln \left(x\right)-2\mathrm{\xi }y+6\mathrm{\eta }x\ln \left(x\right)y^2-\mathrm{I}{x}^2{\mathrm{\eta }}_x+4x{\mathrm{\xi }}_x{y}^3\ln \left(x\right)+2x{\mathrm{\xi }}_{x}y-2x{\mathrm{\eta }}_y{y}^3\ln \left(x\right)-x{\mathrm{\eta }}_{y}y+2\mathrm{\xi }{y}^3+\mathrm{I}{\mathrm{\eta }}_{x,x}{x}^3+\mathrm{\eta }x\right]$

PDEtools:-Library:-UpperBounds(sys[2], [xi,eta](x,y));

$\left\{\mathrm{\eta }=3\,\mathrm{\xi }=2\right\}$

sol[2] := PolynomialSolutions(sys[2]);

${\mathrm{sol}}_2≔\left\{\mathrm{\eta }=y^3\,\mathrm{\xi }=x\right\}$

pdetest(sol[2], sys[2]);

$\left[0\,0\,0\,0\right]$