While talking to our colleagues physicists about modern methods in nonlinear partial differential equations (NPDE) we were often asked for a handbook which could lead into the heart of the matter without the details not directly related to solving. But there were no such books. Evidently the competent authors were reluctant to pay the cost of depth and insight for such a direct approach. And so were we. But the situation made us change our attitude.
In 2005 we were asked to give a series of lectures on NPDE at the Vinh University, Vietnam. The audience was bright and full of commitment but their mathematical background was modest and they had to make an effort to understand the lectures in English. To make their learning more effective, the English lectures of one of the authors (P.P.G.) were followed by classes conducted in Vietnamese by the other one (C.L.V.). As the whole course lasted from 22 to 31 August 2005 (a scarce 24 hours!), there was no time for any extras. We concentrated on the methods which seemed to be most adequate to integrable physical systems