Complex plane representations and stationary states in cubic and quintic resonant systems

Abstract

Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives related to the physics of Bose-Einstein condensates and Anti-de Sitter spacetime) that admit special analytic solutions and an extra conserved quantity. Here, we develop and explore a complex plane representation for these systems modelled on the related cubic Szegő and LLL equations. To demonstrate the power of this representation, we use it to give simple closed form expressions for families of stationary states bifurcating from all individual modes. The conservation laws, the complex plane representation and the stationary states admit furthermore a natural generalization from cubic to quintic nonlinearity. We demonstrate how two concrete quintic PDEs of mathematical physics fit into this framework, and thus directly benefit from the analytic structures we present: the quintic nonlinear Schrödinger equation in a one-dimensional harmonic trap, studied in relation to Bose-Einstein condensates, and the quintic conformally invariant wave equation on a two-sphere, which is of interest for AdS/CFT-correspondence. arXiv:1904.09575v2 [math-ph]

Breathing Modes, Quartic Nonlinearities and Effective Resonant Systems

“…It is a nontrivial fact that the ansatz (46) is consistent with these equation of motion, and yet it is true by virtue of the conservation of B 0 and the identity (36) it implies, as demonstrated in [17]. A key point of the proof is that finite-difference identities (36) imply summation identities for C nmkl adapted to the summation structure in (47).…”

“…We have presented a collection of explicit breathing modes related to the dynamics of Bose-Einstein condensates and Anti-de Sitter spacetimes. While the breathing modes (9) and (12) are standard in the Bose-Einstein literature, the corresponding relativistic breathing modes (17) and (19) are in principle known from the symmetry properties of AdS spacetimes, but we believe our explicit expressions are compact and convenient.…”

“…With β n = (1 + n)(1 + n/G), (36) imposes constraints on the coefficients of the resonant system (38), which are identical to the ones used in [17] to define the 'solvable' class of resonant system. 'Solvability' is understood here in a very restricted sense, namely, as having an explicit family of nontrivial solutions.…”

“…where b(t), a(t) and p(t) are complex-valued functions of time (and the conventions in the above formula differ slightly and inessentially from [17]). The Hamiltonian equations of motion of (38) are…”

“…An explicit bound can be given on the turbulent transfer of energy toward large n modes for the solutions in the ansatz (46). The reader is referred to [17] for detailed derivations of these results. It is worth noting that the above properties were developed in [17] completely in the language of resonant systems of the form (38), without any specific attention to how such features could emerge in resonant system arising as weakly nonlinear approximations to realistic PDEs.…”