On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions

Abstract

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schrödinger type and have recently been obtained in [11] in the context of nonlinear optics. In contrast to the usual nonlinear Schrödinger equation, this new model incorporates the additional effects of self-steepening and partial off-axis variations of the group velocity of the laser pulse. We prove global-in-time existence of the corresponding solution for various choices of parameters, extending earlier results of [2]. In addition, we present a series of careful numerical simulations concerning the (in-)stability of stationary states and the possibility of finite-time blow-up.

On a nonlinear Schrödinger equation for nucleons in one space dimension

“…For = 2, 3 we get in the same way the figures in Figure 2. It can be seen that the higher nonlinearity has a tendency to lead to more compressed peaks as in [1]. But due to the missing scaling invariance of the ground states here, it is difficult to compare them.…”

Numerical study of Zakharov-Kuznetsov equations in two dimensions

“…The latter will be iteratively solved by a Newton-Krylov iteration. This means that we invert the Jacobian via Krylov subspace methods as in [1], here GMRES [26]. We use N x = N y = 2 10 , L x = L y = 10 and Q = 2 e −x 2 −y 2 as initial iterates in all cases.…”

“…We are interested in the 2D generalized Zakharov-Kuznetsov (ZK) equation (1) u t + (u xx + u yy + u p ) x = 0, p = 2, 3, 4.…”

Some variants of the focusing NLS equations

“…It is an open question to get a precise description of the abovementioned "optical shock" phenomenon. In [2], Arbunich, Klein and Sparber have addressed this question, in particular from the numerical point of view. The typical form of (3.12) is…”

“…with the same notations as in (3.10), and for some δ ∈ R d . In [2], the authors show that in dimension d = 2, in the cubic case σ = 1, for partial off-axis variation of the group velocity (k = 1), when the derivative of the nonlinearity is parallel to the regularization (corresponding to Assumption 3.12), solutions to the Cauchy problem associated with (3.13) are global in time (Theorem 6.3). They present numerical evidences, in the case of derivative of the nonlinearity in the orthogonal direction, of blow-up in the absence of off-axis variation, whereas the L ∞ -norm of the computed solution stabilizes when partial off-axis variations are present.…”