The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition

Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

“…The distinction between focusing or defocusing nonlinearity is thus a priori ambiguous. However, in the previous case of Gaussian data in dimension 1, the behaviour of r α (and then that of u α ) has been proven to be sensibly different ( [5,7,18]). Proposition 1.6.…”

“…For such data, the evolution of the solution is given by a single matrix ODE, which can even be simplified in dimension 1 (see [7,2,18]): Proposition 1.5. For any α ∈ C + := {z ∈ C, Re z > 0}, consider the ordinary differential equation It has a unique solution r α ∈ C ∞ (R) with values in (0, ∞).…”

“…For λ > 0, such solutions u α are almost periodic in time (up to a time-depending complex argument), which motivates to call them (and their derivates through the invariants and the scaling effect) breathers. If those solutions are in dimension 1, they can be tensorized in order to find other solutions (also called breathers) in higher dimension, even though they may be not periodic in general (see [18]). However, in higher dimension d ≥ 2, in the general case, the solutions (1.3) are not periodic (and not "almost" periodic) and cannot be put under the form of a tensorization of breathers (1.6) in dimension 1 (already noticed in [5]).…”

Growth of Sobolev norms for coupled lowest Landau level equations

“…It is interesting to observe that the traveling waves defined by (1.14) have a Gaussian decay, which is not common for a Schrödinger-like equation (usually the rate of decay is at most exponential, see e. g. [32]). We can however mention the Schrödinger equation with logarithmic nonlinearity (logNLS) which has Gaussian solitons [12,1,11,19,18] and which possesses several dynamical similarities with (1.3).…”

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

“…For properties of these special solutions see [19], [4], [5], [6], [7], [22], [24]. Besides, the dynamics of the defocusing equation, when + is replaced by − above the nonlinearity, was completely described in the recent work [17].…”