Palestinians say Israeli restraint is needed to allow Abbas, who took office less than a month ago, to consolidate his power, persuade the militants to stop attacks and crack down on them if they refuse.
Critics have previously said Israel's demand that all terror end before any peace moves are carried out gave militants a virtual veto: If they were able to pull off even one attack, all talk of peace would end.
But Israel says it is truly committed to the new peace plan and may have to be more flexible.
The government does not expect that Abbas will be able to stop all the attacks, but they expect him to work tirelessly to end terror, Sharon adviser Raanan Gissin said.
"We say 100 percent effort on dismantling [militant groups], on disarming, on making arrests. Show us that you're making the effort," he said.
Abbas has suggested that he will try persuasion first, arguing in an interview in the Israeli daily Haaretz on Wednesday that his security forces have in any case been decimated by Israel during the fighting.
But he agreed it was important for the Palestinians to stop the attacks and appealed for Israel for patience.
"If we go back to the cycle of reaction and action, that will make it difficult for us to achieve the goal," he said. "It is impossible to achieve 100 percent success in a brief period."
He said Israel needs to end its practice of targeted assassinations, free prisoners and stop demolishing Palestinian houses, which would help gain Palestinian support for the new peace moves "and prevent more suffering."
Sharon's decision not to react to the recent attacks was likely the result of the "overwhelming" US pressure on the leaders to make the road map work, said Gerald Steinberg, a political science professor at Bar Ilan University.
Sharon has worked hard to cultivate a close relationship with Bush, and many believe he does not want to endanger Israel's relationship with its most important ally.
"The full weight and prestige of the United States government is now involved and neither Sharon nor Mahmoud Abbas is willing to say no to Bush," Steinberg said.
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