Senior U.S. officials say they are looking for new approaches to combat piracy off the coast of Somalia. There have been four more ship hijackings since Sunday's dramatic rescue by U.S. forces of an American sea captain who was held hostage by pirates. In all, about 16 ships and more than 250 hostages from various countries are still being held.  

Officials say there are no easy, quick or purely military solutions to the Somali piracy problem. But the top U.S. military officer, Admiral Mike Mullen, told ABC's Good Morning America television program Tuesday the taking and rescue of the American cargo ship captain led him to order a new effort to find ways to effectively deal with the pirates.

"I've actually asked [for] and we've initiated a review on the Joint Staff to look broadly and widely and deeply at the overall strategy," he said.

But so far, some of the new ideas that have been offered by analysts have not been endorsed by national security officials, including a suggestion for a military operation on land in Somalia.

"I think that would be really unfortunate and would most likely backfire," said David Smock, an Africa expert at the U.S. Institute of Peace."I think the pirates would retreat and hide. It would be very hard to have a ground operation that would clear them out. And it would just set the Somali people against the western states, and would only bring closer the relationship between the pirates and the jihadists."

Even Sunday's rescue at sea that resulted in the deaths of three pirates was followed by defiant comments from pirate leaders and jihadists leaders in Somalia.

Admiral Mullen took notice.

"I certainly take their comments afterwards seriously. That said, we're very well prepared to deal with anything like that. And that would certainly be part of our review militarily," he said.

Admiral Mullen and Defense Secretary Robert Gates say there is no purely military solution to the Somali piracy problem. Piracy continues to rise, even as navy ships from 16 nations patrol the area. And Gates said Monday that the problem is "probably going to get worse" until the international community can "get something on land that begins to change the equation" for the "incredible number of poor people" in Somalia, some of whom turn to piracy.

Proposals to help bring stability and economic development to Somalia involve long-term efforts with uncertain outcomes. Efforts by some ship owners to secure their ships have had some success. They have installed barbed wire, removed ladders and changed their routes, among other steps. But there has been resistance to proposals to put armed guards on commercial ships.

Virginia Lunsford at the U.S. Naval Academy has researched the history of piracy around the world and says there may be other ways to reduce the pirates' ability to operate, short of attacking them on land or waiting for the situation in Somalia to change.  (Read her latest article on the subject here)

"In addition to needing to have a base of operations, they typically have some source of support. That support might be in terms of provisioning," she said. "It might be in terms of providing an outlet for the capital that is acquired from the piracy. It might be in the form of local officials who are corrupt, who aid and abet or cover up for them. So another way to get at it is if you can identify some of the explicit sources of support."

Professor Lunsford says despite their defiant statements, Somali groups might not know how to proceed in the wake of a U.S. attack.

"They may be having the discussion about strategically what is it best to do. Do you lay low and hope that ransoms will keep coming in, because that's what's been working for them? Or as a gesture of, sort of, toughness, do you retaliate and say, 'If you don't cooperate, this is what's going to happen to the rest of the hostages," she said.

Experts say the priority for Somali pirate leaders will likely be to protect their lucrative criminal enterprise. Corporations have paid tens of millions of dollars for the safe return of ships and their crews in the area in recent years. Last year, there were 111 piracy incidents in the region, triple the number the previous year. And this year, there have already been nearly 70 incidents.

U.S. officials have stressed the need for increased international cooperation to end the Somali piracy, involving military and civilian efforts.  

In Asia's Straits of Malacca, regional countries have worked together to sharply reduce piracy in recent years, partly with U.S. help. But Somalia's government is in no position to help handle security in the much larger bodies of water off its coasts.

So U.S. officials, including President Barack Obama, say a broader effort is needed to end the piracy.

"To achieve that goal, we're going to have to continue to work with our partners to prevent future attacks. We have to continue to be prepared to confront them when they arise. And we have to ensure that those who commit acts of piracy are held accountable for their crimes," the president said.

But even that is complicated, particularly when some of the pirates are teenagers, like those who took the American cargo ship captain hostage last week, according to U.S. officials.  

Secretary Gates said Monday he is "confident" that he and other officials "will be spending a lot of time in the Situation Room over the next few weeks, trying to figure out what in the world to do about this problem."