Limits of soliton solutions (original) (raw)

Bulletin of the American Mathematical Society, 1992

We announce a detailed investigation of limits of N-soliton solutions of the Korteweg-deVries (KdV) equation as N tends to infinity. Our main results provide new classes of KdV-solutions including in particular new types of soliton-like (reflectionless) solutions. As a byproduct we solve an inverse spectral problem for onedimensional Schrödinger operators and explicitly construct smooth and real-valued potentials that yield a purely absolutely continuous spectrum on the nonnegative real axis and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of the negative real axis.

The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation

Communications on Pure and Applied Mathematics, 2014

We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional.

Soliton Equations and Their Algebro-Geometric Solutions

2008

The discovery of solitary waves of translation goes back to Scott Russell in 1834, and during the remaining part of the 19th century the true nature of these waves remained controversial. It was only with the derivation by Korteweg and de Vries in 1895 of what is now called the Korteweg-de Vries (KdV) equation, that the one-soliton solution and hence the concept of solitary waves was put on a firm basis. 1 An extraordinary series of events took place around 1965 when Kruskal and Zabusky, while analyzing the numerical results of Fermi, Pasta, and Ulam on heat conductivity in solids, discovered that pulselike solitary wave solutions of the KdV equation, for which the name "solitons" was coined, interact elastically. This was followed by the 1967 discovery of Gardner, Greene, Kruskal, and Miura that the inverse scattering method allows one to solve initial value problems for the KdV equation with sufficiently fast-decaying initial data. Soon thereafter, in 1968, Lax found a new explanation of the isospectral nature of KdV solutions using the concept of Lax pairs and introduced a whole hierarchy of KdV equations. Subsequently, in the early 1970s, Zakharov and Shabat (ZS), and Ablowitz, Kaup, Newell, and Segur (AKNS) extended the inverse scattering method to a wide class of nonlinear partial differential equations of relevance in various scientific contexts ranging from nonlinear optics to condensed matter physics and elementary particle physics. In particular, solitons found numerous applications in classical and quantum field theory and in connection with optical communication devices. Another decisive step forward in the development of completely integrable soliton equations was taken around 1974. Prior to that period, inverse spectral 1 With hindsight, though, it is now clear that other researchers, such as Boussinesq, derived the KdV equation and its one-soliton solution prior to 1895, as described in the notes to Section 1.1. 1 To obtain a closed system of differential equations, one has to express F r (µ j ) solely in terms of µ 1 , . . . , µ n and E 0 , . . . , E 2n+1 ; see (1.222) and (1.223).

Soliton equations and the Riemann-Schottky problem

2011

Introduction 1 2 The Baker-Akhiezer functions-General scheme 12 3 Dual Baker-Akhiezer function 17 4 Integrable hierarchies 19 5 Commuting differential and difference operators. 24 6 Proof of Welters' conjecture 27 7 Characterization of the Prym varieties 37 8 Abelian solutions of the soliton equations 45