Multivortex Traveling Waves for the Gross--Pitaevskii Equation and the Adler--Moser Polynomials

From KP-I lump solution to travelling waves of Gross-Pitaevskii equation

“…Up to now, the variational arguments mentioned above have not been able to tackle the higher energy solutions. For these type of solutions, when the speed c is close to zero, in [3,29], the second and third authors applied Lyapunov-Schmidt reduction method to show the existence by gluing suitable copies of the vortex solutions of the Ginzburg-Landau equation. In dimension two, the position of the vortices is determined by the Adler-Moser polynomials; in higher dimensions, the position is determined by a family polynomials, which we called generalized Adler-Moser polynomials.…”

Entire solutions of the magnetic Ginzburg-Landau equation in $\mathbb{R}^4$

“…6 The linearized operator ¿From now on, we will apply standard infinite dimensional Lyapunov-Schmidt reduction. See [19], [7].…”

“…Indeed, for any even dimension not less than 8, the Allen-Cahn equation has a saddle type equation which vanishes on the Simons' cone. Moreover, there is family of solutions whose zero sets are asymptotic to the Simons' cone, See [6,19,18,23] for related results. These solutions are conjectured to be global minimizers of the corresponding Allen-Cahn energy functional, since the Simons' cone is area minimizing when the dimension n ≥ 8.…”

On some configurations of oppositely charged trapped vortices in the plane

“…While a large vein of potential work can be opened by considering three-dimensional settings, we limit our considerations to the 2d case, but involving potentially traveling configurations. There exist works such as those of [35] and more recently [41] which have discussed intriguing algebraic connections including those with the so-called Adler-Moser polynomials (see also references therein). Nevertheless one can envision important, physically relevant variations where the vortices are confined in one direction in the plane, while traveling in the other direction.…”

“…The role of the quantum features arises through the so-called quantum pressure term. This analogy can be utilized to approximate the vortex dynamics and interactions within the GP system by those of point vortices in the fluid setting; for a recent discussion of how to utilize configurations of the latter to prove the existence of steady or co-traveling states in the former, see, e.g., [41]. There is a history of connections between the theory of different types of polynomials and the study of vortices in fluids [35,6,5].…”