Construction of Smooth Chiral Finite-Time Blow-Up Solutions to Calogero–Moser Derivative Nonlinear Schrödinger Equation

Source abstract

We consider the Calogero–Moser derivative nonlinear Schrödinger equation (CM-DNLS), which is a L 2 L^{2} -critical nonlinear Schrödinger equation with explicit solitons, self-duality, and pseudo-conformal symmetry. More importantly, this equation is known to be completely integrable in the Hardy space L + 2 L_{+}^{2} and the solutions in this class are referred to as chiral solutions. A rigorous PDE analysis of this equation with complete integrability was recently initiated by Gérard and Lenzmann [Comm. Pure Appl. Math. 77 (2024), no. 10, 4008–4062, DOI 10.1002/cpa.22203. MR 4814915 ]. Our main result constructs smooth, chiral, and finite energy finite-time blow-up solutions with mass arbitrarily close to that of a soliton, answering the global regularity question for chiral solutions raised by Gérard and Lenzmann. The blow-up rate obtained for these solutions is different from the pseudo-conformal rate. Our proof also gives a construction of a codimension one set of smooth finite energy initial data (but without addressing chirality) leading to the same blow-up dynamics. Our blow-up construction in the Hardy space might also be contrasted with the global well-posedness of the derivative nonlinear Schrödinger equation (DNLS), which is another integrable L 2 L^{2} -critical Schrödinger equation. The overall scheme of our proof is the forward construction of blow-up dynamics with modulation analysis. We begin with developing a linear theory for the near-soliton dynamics. We discover a nontrivial conjugation identity, which unveils a surprising connection from the linearized (CM-DNLS) to the 1D free Schrödinger equation, which is a crucial ingredient for overcoming the difficulties from the nonlocal nonlinearity. Another principal challenge in this work, the slow decay of the soliton, is overcome by introducing a trick of decomposing solutions depending on topologies, which we believe is of independent interest.