582 research outputs found
Asymptotics of rarefaction wave solution to the mKdV equation
Get PDFThe modified Korteweg-de Vries equation on the line is considered. The initial data is the pure step function, i.e. for and for , where are arbitrary real numbers. The goal of this paper is to study the asymptotic behavior of the solution of initial-value problem as , i.e. to study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann-Hilbert problem to the explicitly solvable model forms and show that the solution of the initial-value problem has different asymptotic behavior in different regions of the xt-plane. In the regions and , the main term of asymptotics of the solution is equal to and , respectively. In the region , the asymptotics of the solution tends to . A. Minakov, 2011
Step-initial function to the MKdV equation: Hyper-elliptic long-time asymptotics of the solution
Get PDFThe modified Korteweg-de Vries equation on the line is considered. The initial function is a discontinuous and piece-wise constant step function, i.e. for and for , where are real numbers which satisfy The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as Using the steepest descent method we deform the original oscillatory matrix Riemann-Hilbert problem to explicitly solvable model forms and show that the solution of the initial-value problem has different asymptotic behavior in different regions of the plane. In the regions and the main term of asymptotics of the solution is equal to and , respectively. In the region the asymptotics of the solution takes the form of a modulated hyper-elliptic wave generated by an algebraic curve of genus 2. V. Kotlyarov and A. Minakov, 2012
Modulated elliptic wave and asymptotic solitons in a shock problem to the modified Korteweg-de Vries equation
No full textOn the long time asymptotic behaviour of the modified Korteweg de Vries equation with step-like initial data
Get PDFWe study the long time asymptotic behaviour of the solution q(x,t) of the modified Korteweg de Vries equation (MKdV) with step-like initial datum asymptotic to c_+ at +infinity and to c_- at -infinity. We show that the solution for long times decomposes in the (x,t) plane in three main regions:1. a region where solitons and breathers travel with positive velocities on a constant background c_+;2. an expanding oscillatory region {\color{black} (that generically contains breathers)};3. a region of breathers travelling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the step initial data. The phase shift depends on the solitons, breathers and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers and radiation
Riemann-Hilbert problems and the mKdV equation with step initial data: short-time behavior of solutions and the nonlinear Gibbs-type phenomenon
No full textІсторія поняття досвіду / History of the Concept of Experience
Get PDFThe book is a history of the concept of experience in philosophy. Minakov focuses mainly on Western 19-20th century philosophical movements and their use of the experience concept. Author uses topological method to describe growth of the conseptual content of experience, as well as decline in its use in the end of 20th century
Asymptotics of step-like solutions for the Camassa-Holm equation
No full textWe study the long-time asymptotics of solution of the Cauchy problem for the Camassa-Holm equation with a step-like initial datum. By using the nonlinear steepest descent method and the so-called g-function approach, we show that the Camassa-Holm equation exhibits a rich structure of sharply separated regions in the x,t-half-plane with qualitatively different asymptotics, which can be described in terms of a sum of modulated finite-gap hyperelliptic or elliptic functions and a finite number of solitons
Asymptotic eigenvalue estimates for a Robin problem with a large parameter
No full textRobin problem for the Laplacian in a bounded planar domain with a smooth boundary and a large parameter in the boundary condition is considered. We prove a two-sided three-term asymptotic estimate for the negative eigenvalues. Furthermore, improving the upper bound we get a two term asymptotics in terms of the coupling constant and the maximum of the boundary curvature. © European Mathematical Society
State-Building Politics after the Yugoslav and Soviet Collapse. The Western Balkans and Ukraine in a comparative Perspective. An Introduction
No full textThis issue of the journal Southeastern Europe is dedicated to the comparison of political developments and state-building processes in the so-called Western
Balkans and the Ukrainian case, which is an emblematic post-Soviet Eastern European country. As any other similar exercise, the comparative framework
has its own benefits and limitations. Methodologically and by content, this approach is, in fact, indubitably challenging. On the one hand, comparative studies encourage critical reflections across borders between states or macroregions. On the other, the quantitative analysis of data, which is more frequently applied in these cases, has been replaced, in this issue, by a different approach, preferably based on the qualitative scrutiny of two case-studies, here encoded in the broader geopolitical notion of “South-East Europe”.In this issue we present four articles that cover all above similarities and differences. Their narratives, however, follow a different comparative approach. Two articles, in fact, suggest a binary in-depth analysis between the two peripheral areas under scrutiny. Instead, the other two contributions focus mainly, although not exclusively, on the specific dynamics that are characterizing each of the two macroregions. Our belief is that, in this way, the reader will more effectively grasp both commonalities and peculiarities of these two complex European strategic zones
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