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Self-Similar Singular Solutions to the Nonlinear Schr\"odinger and the Complex Ginzburg-Landau Equations
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We prove the existence of radial self-similar singular solutions for the mass supercritical Nonlinear Schr\"odinger Equation far from the critical regime and, more generally, branches of such solutions for the Complex Ginzburg-Landau Equation. We are also able to control their monotone index (number of monotone intervals). In particular, we prove the existence of monotone radial self-similar singular solutions for the three dimensional cubic Nonlinear Schr\"odinger Equation. The paper combines sharp analytic bounds of the self-similar profile at infinity with computer assisted bounds around zero and their matching at an intermediate value.
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The energy of fractional Allen--Cahn layers in dimension one
The energy of the one-dimensional fractional Allen-Cahn layer is continuous and strictly decreasing in the fractional exponent s, with an explicit pole at s=1/2 and an explicit linear expansion at s=1.
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