Modulation Theory for Self-focusing in the Nonlinear Schrödinger-helmholtz Equation
نویسنده
چکیده
The nonlinear Schrödinger-Helmholtz (SH) equation in N space dimensions with 2σ nonlinear power was proposed as a regularization of the classical nonlinear Schrödinger (NLS) equation. It was shown that the SH equation has a larger regime (1 ≤ σ < 4 N ) of global existence and uniqueness of solutions compared to that of the classical NLS (0 < σ < 2 N ). In the limiting case where the Schrödinger-Helmholtz equation is viewed as a perturbed system of the classical NLS equation, we apply modulation theory to the classical critical case (σ = 1, N = 2) and show that the regularization prevents the formation of singularities of the NLS equation. Our theoretical results are supported by numerical simulations. MSC Classification: 35Q40, 35Q55, 78A60
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