@acer I double checked it. I didn't use lambda[1] in the entire worksheet. 

@dharr Thanks. But now I solved another determinant and this time Maple split 'lambda' into 'real' and 'imaginary' parts in some places and keep 'lambda' or 'conjugate(lambda)' remains same (look at (7)). How to fix it?

Download sol1det.mw

@rlopez because u, ub, and their derivatives are non-abelian i.e. their multiplication are non-commutative. That's why I used bold black dots.

@rlopez I try it as you suggested but got a '0' answer for all the coefficients.

Download compare.mw

@mmcdara when we change "y1:=-1*v^2+1/8*v^4" to "y1:=-1*v^2+1/4*v^4", it shows an error. How to remove it?

pot_mmcdara1.mw 

@ecterrab I removed it. But still not working. 

@tomleslie Can we plot (4)? I tried it by putting "
plot(rhs(%), {_C1 = 1, c = 1, g = 1.5}, x = -10 .. 10);" just after (4). But encountered with "Error, invalid input: rhs received [w(x) = 2*JacobiSN((1/2)*(-2*c+2*(c^2+2*g)^(1/2))^(1/2)*x+_C1, ((c*(c^2+2*g)^(1/2)-c^2-g)*g)^(1/2)/(c*(c^2+2*g)^(1/2)-c^2-g))*g/(g*(-c+(c^2+2*g)^(1/2)))^(1/2)], which is not valid for its 1st argument, expr"

@Rouben Rostamian  When I execute Lineaize.mw on my computer. Maple returns "Error, (in collect) cannot collect 0"

Download Lineaize.mw

@Rouben Rostamian  I got it. But when we use eq (5) into eq (4), maple gives eq (6) after simplification. How can we know the result (eq (6)) is correct?

@Carl Love epsilon derivative raised to power 2 or higher can be removed.

@Carl Love After doing mtaylor, no derivative involve in the final equation. See attachment (where I didn't use mtaylor), there are also derivative terms like diff(epsilon[2], t), etc in equation (6). But I am not clear whether the calculations are going in the right direction or not?

Calculation.mw

@Carl Love as per your suggestion, I got linearize equation. But epsilon[1] and epsilon[2] are functions of x and t. How to do it with epsilon[1]=epsilon[1](x,t) and epsilon[2]=epsilon[2](x,t)?

Calculation.mw  

@Kitonum how to get continuous system from dicrete system by sending h (lattice parameter along x-axis) approaches to zero in maple? i.e., in the attached worksheet, if we send h --> 0, then eq (5) reduces to q_{xt} + (2r)r_{x}=0. How to do that in maple?

CD_new.mw

@Kitonum D₂ is the derivative?

@Carl Love very nice explanation. To understand completely I need more details on "order", "digits" and "degree".