Initial value problem

Quantum mechanics

Norm (philosophy)

Pure mathematics

Mathematical physics

Applied mathematics

Nonlinear Schrödinger equation

Cauchy problem

Economics

Bounded function

Physics

Nonlinear system

Mathematics

Political science

Mathematical analysis

Financial economics

Cauchy distribution

Sobolev space

Schrödinger equation

Derivative (finance)

Inventiones mathematicae

This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces $$H^s({\mathop {{\mathbb {R}}}\nolimits })$$ is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $$H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })$$ with mass strictly less than $$4\pi $$ or general initial conditions in the weighted Sobolev space $$H^{2, 2}({\mathop {{\mathbb {R}}}\nolimits })$$ . In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in $$H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })$$ and that furthermore the $$H^{\frac{1}{2}}$$ norm of the solutions remains globally bounded in time. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.

Global well-posedness for the derivative nonlinear Schrödinger equation