I think I have it. My solution:

eoMkitaevchaindiracnormal = {
   -I c1'[t] == -\[Omega] (c2[t] + ctc2[t]) - 
     v0 Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/2 c1[t],
   -I ctc1'[t] == \[Omega] (c2[t] + ctc2[t]) + 
     v0 Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/2 ctc1[t],
   -I c2'[t] == -\[Omega] (c1[t] - ctc1[t]) - 
     v0 Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/2 c2[t],
   -I ctc2'[t] == \[Omega] (-c1[t] + ctc1[t]) + 
     v0 Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/2 ctc2[t],
   c1[-10] == 1, c2[-10] == 1, ctc1[-10] == 1, ctc2[-10] == 1};
solutionkitaevchaindiracnormal = 
  NDSolve[eoMkitaevchaindiracnormal /. {v0 -> 1, 
     v -> 1, \[Sigma] -> 0.75, x1 -> 1, x2 -> 2, \[Omega] -> 1}, {c1[
     t], c2[t], ctc1[t], ctc2[t]}, {t, -100, 100}];
Plot[Evaluate[{ctc1[t]*c1[t], ctc2[t]*c2[t], 
    ctc1[t]*c1[t] + ctc2[t]*c2[t]} /. 
   solutionkitaevchaindiracnormal], {t, -10, 10}, PlotRange -> All, 
 PlotStyle -> {{Blue, Thick}, {Orange, Thick}, {Black, Dashed, 
    Thick}}]

Thanks for your help.

Now, I have another problem. I want that the absolute value of $c_1, c_1^\dagger,c_2, c_2^\dagger$ is between 0 and 1. Assumptions and Assuming don't work with NDSolve. I know why I can't use Assumptions and Assuming. But I don't know what is the command for NDSolve?

Thanks you for the hint with the double Name. Now Mathematica solve the System. I use ctc1[t} und ConjugateTranspose[c1[t]] and both have the same solution. But I think that Mathematica don't know that ConjugateTranspose[c1[t]] should be $c_1^\dagger$ and >Mathematica interpret c1 and ConjugateTranspose[c1[t]] as operator because if I plot $c_1^\dagger c_1$ then I get

where horizontal axis is the real part and vertical axis is the imaginary part. I think that ConjugateTranspose[c1[t]] is not $c_1^\dagger$. I think that the imaginary part are Zero. I use this Code:

eoMkitaevchaindiracnormal = {
   -I c1'[t] == -\[Omega] (c2[t] + ctc2[t]) + 
     v0 Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/2 c1[t],
   -I ctc1'[t] == \[Omega] (c2[t] + ctc2[t]) + 
     v0 Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/2 ctc1[t],
   -I c2'[t] == -\[Omega] (c1[t] - ctc1[t]) + 
     v0 Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/2 c2[t],
   -I ctc2'[t] == \[Omega] (-c1[t] + ctc1[t]) + 
     v0 Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/2 ctc2[t],
   c1[-0] == 1, c2[-0] == 1, ctc1[-0] == 1, ctc2[-0] == 1};
solutionkitaevchaindiracnormal = 
  NDSolve[eoMkitaevchaindiracnormal /. {v0 -> 0.1, 
     v -> 0, \[Sigma] -> 0.75, x1 -> 1, x2 -> 2, \[Omega] -> 1}, {c1[
     t], c2[t], ctc1[t], ctc2[t]}, {t, -0, 10}];
ParametricPlot[{Re[#], Im[#]} &@
           Evaluate[{ctc1[t]*c1[t]} /. solutionkitaevchaindiracnormal], {t, -0,
            10}, PlotRange -> All, 
          PlotStyle -> {{Blue, Thick}, {Black, Dashed, Thick}}]

When I include

ctc1 = ComplexConjugate[c1];
ctc2 = ComplexConjugate[c2];

then dont solve Mathematica the System.

What is my mistake and who can I describe the operators in Mathematica korrect?

Ok, I have the package FeynCalc installed but I don't find in the list of FeynCalc an other command which solve my system of equations.

Ok, which command can I use for operators?

Thanks for your answer.

I found a lot of nice commands but Mathematica don't work. My code is for the equation are

eoMkitaevchaindiracnormal = {
   -I c1'[t] == -\[Omega] (c2[t] + ConjugateTranspose[c2[t]]) + 
     Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/
      2 c1[t], -I ConjugateTranspose[c1]'[
      t] == \[Omega] (c2[t] + ConjugateTranspose[c2[t]]) + 
     Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/
      2 ConjugateTranspose[c1[t]], -I c2'[
      t] == -\[Omega] (c1[t] - ConjugateTranspose[c1[t]]) + 
     Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/
      2 c2[t], -I ConjugateTranspose[c2]'[
      t] == \[Omega] (-c1[t] + ConjugateTranspose[c1[t]]) + 
     Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/2 ConjugateTranspose[c2[t]], ,
   c1[-100] == 1, c2[-100] == 1, ConjugateTranspose[c1][-100] == 1, 
   ConjugateTranspose[c2][-100] == 1};

Now, I want use NDSolve to solve the equation and use this code

solutionkitaevchaindiracnormal = 
         NDSolve[eoMkitaevchaindiracnormal /. {v0 -> 1, 
            v -> 1, \[Sigma] -> 0.75, x1 -> 1, x2 -> 2, \[Omega] -> 1}, {c1[
            t], c2[t], ConjugateTranspose[c1][t], 
           ConjugateTranspose[c2][t]}, {t, -100, 100}];

After the second step Mathematics give me this Output:

NDSolve::deqn: Equation or list of equations expected instead of Null in the first argument {-I c1^\[Prime](t)==1/2 E^(-0.888889 Plus[<<2>>]^2) c1(t)-c2(t)-c2(t)^\[ConjugateTranspose],-I ((c1^\[ConjugateTranspose])^\[Prime])\[InvisibleApplication](t)==c2(t)+1/2 E^(-0.888889 Plus[<<2>>]^2) c1(t)^\[ConjugateTranspose]+c2(t)^\[ConjugateTranspose],-I c2^\[Prime](t)==-c1(t)+1/2 E^(-0.888889 Plus[<<2>>]^2) c2(t)+c1(t)^\[ConjugateTranspose],-I ((c2^\[ConjugateTranspose])^\[Prime])\[InvisibleApplication](t)==-c1(t)+c1(t)^\[ConjugateTranspose]+1/2 E^(-0.888889 Plus[<<2>>]^2) c2(t)^\[ConjugateTranspose],Null,c1(-100)==1,c2(-100)==1,(c1^\[ConjugateTranspose])\[InvisibleApplication](-100)==1,(c2^\[ConjugateTranspose])\[InvisibleApplication](-100)==1}. >>

What is my mistake? Can I use ConjugateTranspose too describe a creation operator or I have use an other command ?