Solve an equation in two variables containing symbolic constants

Question

I've been struggling to solve equations in two variables containing constant symbolic parameters. Take for example the function in two variables defined by:

F[x_,y_]:=Sqrt[2 t1^2 + t3^2 + 4 t1 t3 Cos[(3 x)/2] Cos[(Sqrt[3] y)/2] + 2 t1^2 Cos[Sqrt[3] y]]

My goal is to solve $$F(x,y)=0$$ so I want to find the coordinates of the points in the 2D xy plane where the function is 0. These coordinates will be given by an expression containing the parameters $t_{1}$, $t_{3}$. I can find some solutions by guessing, for example: $$(x,y)=(0,\frac{2}{\sqrt{3}}\arccos{(-\frac{t_{3}}{2t_{1}})}) $$

or $$(x,y)=(0,-\frac{2}{\sqrt{3}}\arccos{(-\frac{t_{3}}{2t_{1}})}) $$

I would like to find these solutions (and the others) in Mathematica,so I've tried:

Solve[F[x, y] == 0, {x, y}]

or

Reduce[F[x, y] == 0, {x, y}]

However my problem is that Mathematica gives me an expression where the solution for x contains y or viceversa, but I would like the solutions separate (so the solution for x shall not contain y and viceversa, as in the two I wrote above). Also, sometimes the parameters $t_{1}$,$t_{3}$ are treated as variables by Mathematica, and a solution is provided also for them, which I don't want.

This is a problem I had in general for these kind of equations where I solve for 2 variables and I also have additional constant parameters. So how can I get the solution for x and y (without them been coupled) in terms of the constant parameters?

Answer 1

Solve for Reals with Method->Reduce like @BobHanlon proposes.

Plot solutions for x and y in dependence of t1 and t3, or get values at definite t1,t3.

F[x_, y_] = 
  Sqrt[2 t1^2 + t3^2 + 4 t1 t3 Cos[(3 x)/2] Cos[(Sqrt[3] y)/2] + 
    2 t1^2 Cos[Sqrt[3] y]]

sol3[c1_, c2_, t1_, t3_] = 
 Solve[F[x, y] == 0, {x, y}, Reals, Method -> Reduce] /. {C[1] -> c1, 
   C[2] -> c2}

Plot3D[Evaluate[y /. sol3[0, 0, t1, t3] // Simplify // Union], {t1, 0,
   1}, {t3, 0, 1}, PlotStyle -> {Red, Green, Blue, Magenta}, 
 PlotRange -> All]

{x, y} /. sol3[0, 0, 1/3, 1/2] // TableForm