We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on 𝕋 , also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces H^s_0(𝕋, ℝ) , s > -1/2 , to the scale of weighted ℓ ^2- sequence spaces, 𝔥^s +1/2_r,0(ℕ, ℂ) , s >-1/2 . As an application we show that for any -1/2

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space H-s(T, R) for any s >= 0. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.

In this paper we investigate some number theoretic properties of the frequencies of the Korteweg-de Vries equation on the torus, relevant for the stability of finite gap solutions.

The aim of this chapter is to present a brief introduction to the theory of ordinary differential equations. The main focus is on systems of linear differential equations of first order in $$\mathbb {R}^n$$ with constant coefficients. They can be solved by the means of linear algebra. Hence this chapter is an application of what we have learned so far to a topic in the field of analysis.

We prove that the Benjamin–Ono equation on the torus is globally in time well-posed in the Sobolev space H^s(𝕋,ℝ) for any s > - 1/2 and ill-posed for s ≤ - 1/2. Hence the critical Sobolev exponent s_c=-1/2 of the Benjamin–Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in H^s(𝕋,ℝ) for any s > - 1/2. Novel conservation laws and a nonlinear Fourier transform on H^s(𝕋,ℝ) with s > - 1/2 are key ingredients into the proofs of these results.

The aim of this chapter is to discuss the basic operations on matrices, to introduce the notions of linear (in)dependence of elements in $$\mathbb {R}^k$$ , of a basis of $$\mathbb {R}^k$$ and of coordinates of an element in $$\mathbb {R}^k$$ with respect to a basis. Furthermore, we extend the notion of the determinant of $$2\times 2$$ matrices, introduced at the end of Sect. 1.2 , to square matrices of arbitrary dimension and characterize invertible square matrices as being those with nonvanishing determinant.

In this chapter we introduce the notion of a vector space over $$\mathbb {R}$$ or $$\mathbb {C}$$ and of maps between such spaces which respect to their structure. Maps of this type are referred to as linear maps.

The textbook is designed for first semester students who need to learn linear algebra as well as some computational skills.

In this chapter we introduce the important notions of eigenvalue and eigenvector of a linear map $$f\colon V \to V$$ on a vector space V of finite dimension. Since the case where V is a $$\mathbb {C}$$ -vector space is somewhat simpler, we first treat this case.

So far we have worked with real numbers and used that they are ordered and can be added and multiplied, tacitly assuming that addition and multiplication satisfy the classical computational rules, i.e., that these operations are commutative, associative, $$\ldots $$ . It turns out that for many reasons, it is necessary to consider an extension of the set $$\mathbb {R}$$ of real numbers. These more general numbers are referred to as complex numbers and the set of them is denoted by $$\mathbb {C}$$ .

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H-0(s) (T, R) for any s > -1/2 where H-0(s) (T, R) denotes the subspace of the Sobolev space H-s(T, R) of elements with mean 0. As an application we show that for any -1/2 < s < 0, the flow map of the Benjamin-Ono equation S-0(t) : H-0(s)(T, R) -> H-0(s) (T, R) is nowhere locally uniformly continuous in a neighborhood of zero in H-0(s)(T, R). (c) 2021 Published by Elsevier Ltd.

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not C∞-smooth.

We prove an infinite-dimensional version of the Arnold-Liouville theorem for integrable nonlinear PDEs. In a case study we consider the focusing NLS equation with periodic boundary conditions.

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localization properties of the torsion function are obtained under that hypothesis. An example is analyzed in detail.