Waves, singularities, and nonlinear dynamics.

Finite-time blow-up is studied for the one-dimensional nonlinear Schrödinger equation with combined nonlinearities. Two families of sufficient conditions are obtained: negative-energy conditions for a triple nonlinearity and for convergent infinite sums, including an exponential nonlinearity; and a positive-energy condition for a double nonlinearity with defocusing–focusing coefficients. Examples of initial data satisfying both kinds of conditions are provided.

Consider \[ i u_t + u_{xx} + \mathcal{N}(u)u=0, \qquad x,t\in\mathbb{R}, \] where \[ \mathcal{N}(u)=\sum d_k |u|^{\alpha_k}, \qquad \alpha_k>0. \] For weighted initial data in a subclass of \(H^1(\mathbb{R})\), local well-posedness is established for arbitrary positive exponents using the Cazenave–Naumkin approach rather than Strichartz estimates. A pseudo-conformal transformation gives global solutions for quadratic-phase data, together with scattering in \(H^1(\mathbb{R})\) when the phase \(e^{ib|x|^2}\) has sufficiently large \(b>0\). Infinite sums permit nonlinearities such as \(e^{\gamma |u|^k}u\), as well as sine- and cosine-type nonlinearities. Numerical experiments cover double-power and exponential examples and show that the ground state does not generally form a sharp boundary between scattering and finite-time blow-up.

The one-dimensional focusing stochastic nonlinear Schrödinger equation is considered in the \(L^2\)-critical and supercritical regimes, with multiplicative Stratonovich noise driven by a Wiener process that is white in time and colored in space. The mass, represented by the \(L^2\)-norm, is conserved, whereas the Hamiltonian energy is not. Theory and computation examine how several spatial-correlation kernels, discretization parameters, and numerical schemes affect the energy. Numerical experiments then estimate blow-up-versus-scattering probabilities and indicate that spatially correlated perturbations do not materially change the blow-up dynamics, apart from shifting the blow-up center.

For the one-dimensional nonlinear Schrödinger equation with power nonlinearity \[ |u|^\alpha u, \qquad \alpha>0, \] two methods for local well-posedness are compared. The first uses Strichartz estimates with initial data in \(L^2\) or \(H^1\); in the \(H^1\) setting it typically applies when \(\alpha\geq 1\). The second uses weighted estimates that commute with derivatives, together with an infimum condition on the initial data. This weighted approach covers the full range \(0<\alpha<\infty\) and can treat sums of nonlinearities. It is applied to finitely many combined nonlinear terms, including physically relevant models such as those arising in laser optics, where the absence of scaling invariance makes the first approach difficult to use.