Solve the transcendental equation

By "I know the v-i curve of this equation" you just mean that you have a picture of the plot(s), nothing more?

I would suggest that you trace the plot as precise as possible, creating vectors of corresponding x- and y -values (or v- and i-values in your case) and hen you may use Prime to do a fit determining a best guess for the missing parameters.

You can digitize the given plot(s) manually or use a software (plot digitizer, tracer) to do so. I have no experience so I can't recommend a specific software.

But I am in no way sure that this would work out satisfactory as your v-i-function seems to be dependent on 10 parameters (Vj, Vt, Rs, TM, RM0, i, TMo0, tau, ISE and IS) and the parameter Tj which is mentioned in the plots is not even part of them!?

You only mention 6 of these parameters - does this mean that the other four are known?

Its also unclear to me which parameter/argument in th equations you provide would combine the v- and i-values to form the plot shown.

Nonetheless, fitting a function with as much as six free to choose parameters sure also has a lot of degrees of  freedoms and I would estimate that it will depend extremely on good guess values whether the results are desired and usable.

EDIT: Additionally I just wonder if the "i" which is used in the first equation for "v" is equivalent to the expression for "i" given in the second equation??

Looks to me that you have quite something to clarify. Digitize your plots (at least one trace of them), set up a worksheet and state clearly the dependencies.