1. There was no direct order for Bernadotte with Davout’s orders. Davout’s actions support this, with his question and later absence of any accusations of ‘abandonment.’ His direct orders were for Bernadotte to march to Dornburg, which he did badly.

2. Napoleon didn’t fault Bernadotte for not marching with Davout. He faulted him for his ‘false march’, failing to arrive in Dornburg and failing to come into action.

I have a different opinon about that.

The Prince de Ponte-Corovo did not fail badly on his march to Dornburg and then later to Apolda, two indipendent testimonial - Dupont as well as Belliard comfrim the difficult terrain.

Also - initially the battle of Jena was designed to happen on two days, so for the 15th both Bernadotte as well as Davout would be in an excellent position to take part and come into action.

On the 14th then, Napoléon did beat the four different Prussian army bodys piecemeal, and then at 16:00 the issue was decided - and the battle had shifted much more to the west than anticipiated and ended in one day.

Indeed Bernadotte was placed superbly for the designated pursuite, and he cut of the line of retreat from Prussian units from Auerstedt, which had to make detours - orientate themselves differently or to get captured.

All this and much more is competently discussed in

Stephane Béraud : La révolution militaire napléonienne - tome 2 Les Batailles, 2013

Along with his volume 1 - one of the best works ever published on Napoleonic Warfare.