The researchers from the Center for Mathematical Modeling at Universidad de Chile (CMM) Michał Kowalczyk and Claudio Muñoz, together with Yvan Martel, from École Polytechnique in France, managed to solve a problem that had stayed unsolved for nearly 40 years.
The dilemma, related to Nonlinear Partial Differential Equations, sought to determine if the perturbations of the so called “kink” ¬—a special solution to ‘phi 4’, classic equation of the quantum field theory— must converge when time reaches to infinite.
The answer is a key advancement in the area of disperse equations, those that explain phenomena such as the waves in the ocean, light waves and field physics, among others.
“We proved that, in the case where perturbations are odd, all of them must converge to zero “locally” in space”, explains Muñoz.
They had been working for more than two years solving this problem, remembers Kowalczyk, polish mathematician expat in Chile since 2003: “Everything started when I went as part of a study group to France in 2014. I went there to have a conversation with Yvan and I thought that he would give me the answer, because I was sure that it was already solved”.
But Martel answered that the conjecture was unanswered since the 80’s. The french had been director of thesis for Muñoz and they had known each other since 2006. When the polish visited France, the chilean was working at Université Paris-Sud and was conscious of the immensity of the challenge: “Many people working on Mathematical Physics had tried to give different answers to this problem, with different degrees of rigorousness. Our result is the first hundred per cent strict solution to that question”.
The solution allowed them to be the first researchers working in Chile to publish in the prestigious Journal of the American Mathematical Society, considered one of the three most important scientific magazines in the field of Mathematics and an emblem of the American Society of Mathematics.
“We solved in a partial way the conjecture that equation phi 4 doesn’t have “wobbling kinks”, or periodic solutions in time and located in space around a kink” adds Muñoz. “And the most probable is that perturbations of the “wobbling kink” kind do not exist for phi 4″.
Now, the challenge that face the researchers has to do with answering the missing case, the one of the paired data. With this, they´d give a solution to the general problem, says Claudio Muñoz: “That case has an additional component, called in the area “resonance” that causes its demonstration to be even more complex and defiant than the odd case that we have just solved”. Notwithstanding the obstacles, they have had advancements and the team has produced an intermediate result that has been sent to a scientific magazine. In this article, they tell they have found which is the fundamental problematic of the paired case. Now, they have to solve it.
