President Donald Trump (R) shakes hands with North Korea's leader Kim Jong Un as they meet at the start of their historic US-North Korea summit, at the Capella Hotel on Sentosa island in Singapore on June 12, 2018.

President Donald Trump praised North Korea's Kim Jong Un as "very open" and "terrific" Monday and announced he likely will hold a second summit with Kim "quite soon," striking a far less ominous note at the United Nations about the nuclear threat from the North.

Twelve months after Trump stood at the rostrum of the U.N. General Assembly and derided Kim as "Rocket Man," the push to denuclearize the Korean Peninsula is a work in progress, although fears of war have given way to dreams of rapprochement. The president's bellicose denunciations of Pyongyang have largely given way to hopeful notes, as Trump praised Kim as "very open" and "terrific."

"It was a different world," Trump said Monday of his one-time moniker for the North Korean leader. "That was a dangerous time. This is one year later, a much different time."

He added that preparations are underway by Secretary of State Mike Pompeo for a second presidential meeting with Kim "quite soon."

The meeting would come despite U.S. officials assessing that the North has not followed through on its commitments to take steps toward denuclearization. Pompeo defended Trump's decision to seek another meeting despite the slow progress.

"We've been at this the other way an awfully long time and failed," he said.

He added: "We tried to do details. We tried to do step for step. We tried to do trade for trade. Each of those failed."

Pompeo said, "We're bringing the two senior leaders, the individuals who can actually make the decisions that will move this process forward," in hopes they can make a breakthrough.

Trump arrived at the U.N. on Monday morning for a meeting on the global drug trade, ahead of a sit-down with South Korean President Moon Jae-in, who delivered a personal message to Trump from Kim after their inter-Korean talks last week.

"You are the only person who can solve this problem," Moon said to Trump, relaying Kim's words.

Trump, for his part, said "We are in no rush. We are in no hurry" to bring about a nuclear agreement. U.S. officials are insisting that economic sanctions remain in place against the North until it eliminates its nuclear program.

Trump and Moon are expected to sign a new version of the U.S.-South Korean trade agreement, one of Trump's first successes in his effort to renegotiate trade deals on more favorable terms for the U.S. Even so, some U.S. officials worry that South Korea's eagerness to restore relations with the North could reduce sanctions pressure on Kim's government, hampering efforts to negotiate a nuclear accord.

The nuclear threat was on the agenda at Trump's first meeting in New York, a dinner with Japanese Prime Minister Shinzo Abe in Manhattan on Sunday night. Abe stands first among world leaders in cultivating a close relationship with the president through displays of flattery that he has used to advance his efforts to influence the unpredictable American leader.

"We have our eyes wide open," Pompeo told NBC's "Meet the Press" on Sunday. "There is a long ways to go to get Chairman Kim to live up to the commitment that he made to President Trump and, indeed, to the demands of the world in the U.N. Security Council resolutions to get him to fully denuclearize."

Trump, redoubling his commitment to "America First" on the most global of stages, will stress his dedication to the primacy of U.S. interests while competing with Western allies for an advantage on trade and shining a spotlight on the threat that he says Iran poses to the Middle East and beyond.

Scores of world leaders, even those representing America's closest friends, remain wary of Trump. In the 12 months since his last visit to the U.N., the Republican president has jolted the global status quo by pulling out of the Iran nuclear deal, starting trade conflicts with China and the West and embracing Russia's Vladimir Putin even as the investigation into the U.S. president's ties to Moscow moves closer to the Oval Office.

Long critical of the United Nations, Trump delivered a warning shot ahead of his arrival by declaring that the world body had "not lived up to" its potential.

"It's always been surprising to me that more things aren't resolved," Trump said in a weekend video message, "because you have all of these countries getting together in one location but it doesn't seem to get there. I think it will."

Nikki Haley, the U.S. ambassador to the United Nations, told reporters in a preview of Trump's visit that the president's focus "will be very much on the United States," its role and the relations it wants to build.

"He is looking forward to talking about foreign policy successes the United States has had over the past year and where we're going to go from here," she said. "He wants to talk about protecting U.S. sovereignty," while building relationships with nations that "share those values."

In his four-day visit to New York, Trump will deliver major speeches and meet with representatives of a world order that he has so often upended in the past year. On Monday he participated in a Global Call to Action on the World Drug Problem and later was to meet with Egyptian President Abdel-Fattah el-Sissi and French President Emmanuel Macron, in addition to Moon.

Trump's address to the General Assembly comes Tuesday, and on Wednesday he will for the first time chair the Security Council, with the stated topic of non-proliferation of nuclear, chemical and biological weapons. The subject initially was to have been Iran, but that could have allowed Iranian President Hassan Rouhani to attend, creating a potentially awkward situation for the U.S. leader.

Aides say the president will also use the session to discuss North Korea and other proliferation issues. While Trump is not seeking a meeting with Rouhani, he is open to talking with the Iranian leader if Rouhani requests one, administration officials said.

In meetings with European leaders as well as during the Security Council session, Trump plans to try to make the case that global companies are cutting ties with Iran ahead of the reimposition in five weeks of tough sanctions against Tehran. The penalties are a result of Trump's decision to withdraw the U.S. from the 2015 Iran nuclear deal.

Trump at the time cited Iran's role as a malign force in the region, particularly its support of terrorist groups, but also its involvement in Syria. U.S. officials say their priority for the region now is removing Iranian forces from Syria.

Trump is also expected to deliver a fresh warning to Syria's Bashar al-Assad that the use of chemical weapons against civilians in the major rebel stronghold of Idlib would have serious repercussions.