Abstract: This talk deals with several equations of Mathematical Physics as mKdV, Camassa-Holm and its generalizations, Hunter-Saxton, Burgers, systems of conservation laws, sin-Gordon, semilinear wave equations, etc. They originate from Physics, but we propose here their investigation via purely mathematical methods in the frames of the University courses (Lebesgue integral, Fourier transform, Schwartz distributions). Therefore, the talk is addresses to a broader audience including graduate students, Ph.D. students, mathematicians, physicists, engineers and specialists in the domain of PDE. Certainly, there are monographs on the subject based on rather complicated and difficult methods that make the readers acceptance hard-especially for beginners or non specialists. We propose a short survey on Jacobi’s and Legendre elliptic functions and illustrate them by traveling wave solutions and their interaction. We discuss some examples from physics – interaction of fluxons and antifluxons. By using the appropriate modifications of the method of characteristics we study the generalized Cauchy problem for the Hunter-Saxton equation and for the conservation laws equation (existence, uniqueness). We estimate the life span of the continuous generalized solutions and prove that in some cases they are Lipschitz. Moreover, in some cases we construct ä-shocks avoiding Colombeau algebras and the parabolic regularization.

Brief Biography of the Speaker:
P. Popivanov graduated at the Faculty of Mathematics of the Sofia University "St. Kliment Ohridski" in 1969. In the period 1970-1973 he was PhD student at the Moscow State University "Lomonosov" to the Russian mathematician Yu.V.Egorov – eminent specialist in PDE. He defended his PhD thesis in Moscow in 1973 and his second doctor degree at the Sofia University "St. Kliment Ohridski" in 1986. Since 1969 he works at the Institute of Mathematics and Informatics of BAS: Assoc. Prof. (1979), Full Prof. (1988), Corresponding Member of BAS (1995), Full Member of BAS (2003). He is head of Differential Equations Section at the Institute of Mathematics and Informatics since 1989. He was Chairman of the Scientific Council of the same Institute in the period 1995-2008; and since 2004 he is member of the Board of BAS. P. Popivanov is actively working in the domain of Partial Differential Equations. He is the author of more than 126 scientific papers, 46 of them published in journals having IF; 46 referee reports in Bulgaria, Italy and Germany for PhD and Doctoral theses, habilitations, etc.; 3 monographs – 1 published in Akademie Verlag, Berlin (jointly with D. Palagachev), 1 published in Wiley-VCH (jointly with T. Gramchev), and one in Bulgaria as well as two manuals on Differential Equations. He has been invited as Visiting Professor, for giving seminars and for participation in congresses and conferences in the following scientific centres: Universite Paris-11 (Orsay), Ecole Polytechnique (Paris), Universite Paul Sabatier (Toulouse), Universite de Rennes - France; the Universities of Bologna, Trieste, Torino, Pisa, Ferrara, Bari, Cagliari, Catania, Messina – Italy; Warsaw University – Poland; Moscow State University "Lomonosov", "Steklov" Institute (Moscow) – Russia; Institute "Weierstrass" – Berlin and the Universities in Potsdam, Kemnitz, Holzhau, Clausthal – Germany; the Universities of Lund, Vaxjo, Linkoping – Sweden; the Universities of Chow, Tsukuba, Tokyo, Hiroshima, Kyoto, Nagoya, Osaka – Japan; Weizmann Institute, Technion and Ariel Universities – Israel; Belgrade University – Serbia, Ioannina University – Greece, etc. P. Popivanov is Doctor Honoris Causae of Rousse University "Angel Kanchev".