Simultaneous solving of two equations for extraction of one variable

Questions
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1

I have the following system of equations

exp1 = c \partial_{\chi} u_B + 2 u_B (u_B h_1 + v_B h_2) + \nu v_B + \beta v_B (u_B^2 + v_B^2) + \alpha \partial_{\chi}^2 v_B == 0\\ exp2 = c \partial_{\chi} v_B + 2 v_B (u_B h_1 + v_B h_2) - \nu u_B - \beta u_B (u_B^2 + v_B^2) - \alpha \partial_{\chi}^2 u_B == 0

where I have

h_1=\text{TrigToExp}[\tanh (k x)];\\h_2=\text{TrigToExp}[\text{sech}(k x)];\\ u_B=\sqrt{\gamma +\mu } \text{TrigToExp}[\tanh (k x)]\\ v_B=\sqrt{\mu -3 \gamma } \text{TrigToExp}[\text{sech}(k x)]
2

Your equation is linear in c but it is trascendental in k. You get a closed-form result for example if you solve first for c in one of the equation, replace into the other, and give specific values for the parameters:

Solve[exp2 /. 
   Solve[exp1, c] /. {\[Mu] -> 4, \[Gamma] -> 0, \[Alpha] -> 
    1, \[Beta] -> 1, \[Nu] -> 1}, x]

It is not clear to me what you mean with the third argument in Solve[{exp1,exp2},{c},{k}]

3

But what if I want to solve for c for any values of the parameters? This system does not allow me to do it?

4

Did you try SolveAlways?