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I desperately need to read this paper, before meeting a would-be supervisor but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, Rudin's Analysis, Rabenstein's Ordinary Differential Equations, etc. not even one sentence of the paper is readable to me, so I even don't know what area of mathematics is it about!

What is the necessary prerequisite knowledge to be able to understand the mentioned paper and the best introductory books written about? Thank you!

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    $\begingroup$ I'm unsure what you mean by "I need to read" --- I presume this is because of some research project given to you by an advisor? I presume they would then be the goto person for background info, tailored on what courses you have followed. In particular, that person would at least be able to explain "what area of mathematics" this is about. Here you are unlikely to find a helpful response since we do not know your background. $\endgroup$ – Carlo Beenakker Feb 23 '19 at 14:50
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A couple hints:

  • Equation (1.1) is an equation involving partial derivatives (that's what the subscripts denote), so you're in the area of PDE.

  • Look up the Mathematics Subject Classification codes on the first page. That is another way to identify the general area of mathematics covered by a paper.

  • "A detailed discussion of the subject and corresponding references can be found in the survey [9]." That's probably a good paper to obtain and look at.

  • Don't use "et al" when talking about a paper with just two or three authors. It's impolite to the other authors.

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OK, with the motivation given by the OP in the edited question here is how I would approach the job interview with your would-be advisor. The worst thing you could do is to fake advanced knowledge when you lack basic understanding. I am sure you will make a much better impression if you admit that the paper you were asked to study is far from your expertise, but that you have made an effort to understand the basics you would need to pursue that line of research. So the first step would be to learn what is Darboux integrability. This book chapter could be a good entry point: Darboux integrability, but I'm sure you'll find other entry points if you search the web.

The topic of the paper you were asked to study is to find a way to discretize this special nonlinear differential equation referred to as a Laine equation, without spoiling the existence of the integrals of motion that are the defining property of Darboux integrability. At this point your would-be advisor might ask: "Is the Laine equation Darboux integrable", and it would be great if you could then go to the blackboard and show that it is.

I'm pretty sure that would be sufficient for the interview: you have shown you can place a new topic in its proper context and that you can do the background reading you need in order to enter this new field. If you want to go further, and you have the time, you could see if you can make sense of the discretization procedure advanced in the paper. But that would be optional, I think.

Good luck!

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  • $\begingroup$ Do you really think that it is useful to encourage this kind of questions on this site by answering them? $\endgroup$ – Alexandre Eremenko Feb 23 '19 at 21:44
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    $\begingroup$ I was persuaded by a personal comment of the OP, now removed. $\endgroup$ – Carlo Beenakker Feb 23 '19 at 21:56
  • $\begingroup$ @AlexandreEremenko, what about this?! $\endgroup$ – user135767 Feb 24 '19 at 11:04
  • $\begingroup$ @72D: I was hesitating when answering this previous question. So when a similar question is asked again I decided to seek for an advise of a more experienced user. $\endgroup$ – Alexandre Eremenko Feb 24 '19 at 14:17