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

Abstract

We consider the logarithmic Schrödinger equation (logNLS) in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson -a time-independent Gaussian function -is an orbitally stable solution. In this paper, we construct multi-solitons (or multi-Gaussons) for logNLS, with estimates in H 1 ∩ F (H 1 ). We also construct solutions to logNLS behaving (in L 2 ) like a sum of N Gaussian solutions with different speeds (which we call multi-gaussian). In both cases, the convergence (as t → ∞) is faster than exponential. We also prove a rigidity result on these multi-gaussians and multi-solitons, showing that they are the only ones with such a convergence.

On multi-solitons for coupled Lowest Landau Level equations

“…Such a phenomenon is in contrast with NLS, where the solitons have an exponential decay and where the speed of convergence depends on the frequencies of the solitons [6]. The same rate of decay as in (1.13) is obtained in [12] where multi-Gaussian solutions are constructed for the Schrödinger equation with logarithmic nonlinearity (logNLS). Another interesting similarity with the results in [12], is that the convergence to the multi-soliton holds in weighted Sobolev spaces (namely in H 1 ∩ F(H 1 )).…”

“…The same rate of decay as in (1.13) is obtained in [12] where multi-Gaussian solutions are constructed for the Schrödinger equation with logarithmic nonlinearity (logNLS). Another interesting similarity with the results in [12], is that the convergence to the multi-soliton holds in weighted Sobolev spaces (namely in H 1 ∩ F(H 1 )). In the present case, one can even upgrade to exponential weights, and this is due to the absence of linear part in the equation (1.2) (see Remark 1.4 for the case of LLL with a linear part).…”

“…The construction of multi-solitons for (1.2) relies on classical arguments, including backwards in time integration and energy estimates. We refer to [22,6,7,12,11] where these methods were used. The situation here is very favorable since in the space E, any L p norm (p ≥ 2) can be controlled (see (1.27)), namely…”

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).…”

Around plane waves solutions of the Schr{ö}dinger-Langevin equation

“…A lot of properties of these two equations are already known on the whole space R d . The mathematical study of equation (2) for the focusing case λ < 0 goes back to [9], and the global existence of solutions is now well understood (see [10] and [16]), as well as their qualitative behaviors (see for instance [1], [19] or [20]). On the other hand, the defocusing case λ > 0 for (2) has received some recent attention, in particular from the work [8] which established the global existence and the uniqueness of solutions as well as their asymptotic behavior.…”