If a person promises something in dua like "i will not do such and such sin again" and does not vow in the name of Allah, and then breaks the promise then will he have to expiate(kaffara) for that ?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

If the expression that is mentioned in the question did not include uttering an oath or a promise to Allaah, then one is not obliged to expiate for breaking this, but he is obliged to repent from the sin which he had committed.

However, if this expression included giving a promise to Allaah like saying for example: “I promise Allaah not to return to this sin again”, and then he broke his promise and he did not fulfill it, then he is obliged to repent and to expiate for breaking an oath.

For more benefit on the ruling of making a promise to Allaah and how to expiate, please refer to Fatwa 102811.

Finally, it should be noted that a Muslim should stop sinning even though he did not make a promise to Allaah to stop sinning, and if one made a promise, then it is more confirmed in his regard to abandon the sin.

Allaah Knows best.