A series of talks about non-linear waves, energy supercritical problems, solitons, defocusing Non-Linear Schrödinger equation and shock waves.
Tag - Solitons
The Kademtsev-Petviashvily (KP) equation is a famous evolution equation with soliton solutions. It was discovered by M.Sato and the Kyoto school that the KP equation can be regarded as a part of a countable system of compatible evolution equations, which is called today the KP hierarchy. The observation allowed the researchers to discover many new examples of soliton type hierarchies and to study them with methods of mathematical physics, algebraic geometry and representation theory. In the talk we will describe the explicit construction of polynomial tau-functions of the KP, BKP hierarchies through their generating functions. The method uses the tools of representation theory and properties of symmetric functions.
The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev-Petviashvili equation.
The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin-Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.
An LMS online lecture course in solitons.
The lectures will highlight some recent work on solvable models of topological solitons. The first involves generalisations of the U(1) Abelian-Higgs model whose integrability is intimately related to the geometry of constant curvature Riemann surfaces. The second piece of work is a study of magnetic skyrmions in chiral magnets. Recently a family of soluble models for magnetic skyrmions in chiral magnets was introduced. The energy functional for these models is bounded below by the topological charge, configurations which attain this bound solve first-order equations. The explicit solutions of these first-order equations are given in terms of arbitrary holomorphic functions. Finally I will explain how this model can be interpreted as a gauged non-linear sigma model.
Lecture 1: A primer on solitons. I will introduce the concept of a topological soliton through two prototypical examples, the φ4 and Sine-Gordon models in 1+1 dimensions. Next, we will meet Derrick's theorem and learn why solitons are hard to construct in higher dimensions. Finally, we will meet some examples of higher-dimensional models possessing soliton solutions.
Lecture 2: Solitons in chiral magnets. We will meet a specific model of 2-dimensional chiral magnetic systems which admits soliton solutions. For a special potential term exact, degree 1, skyrmion solutions can be constructed. This leads up to meeting a critically coupled version of the model where there is a whole zoo of analyitic skyrmion solutions.
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