Can anyone assist with asymptotic derivation of phase shift of the two-soliton solution for the KdV equation?

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Can anyone help with the solution to the KdV equation using Hirota's bilinear method?

I am solving the KdV equation using Hirota's method. I used the perturbation series to solve for the one-, two-, three-, and N-soliton solutions but I am having some challenges in the substitution...

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Dmitry Kovriguine

Minimize by phase parameters the difference between the energy of approximate colliding solitaries and their exact asymptotics

Da-jun Zhang

You may go to the paper Solutions to the modified Korteweg–de Vries equation (Reviews in Mathematical Physics, DOI: 10.1142/S0129055X14300064) where you can find examples of asymptotic analysis for two solitons interaction.