Jyh-Hao Lee

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

Abstract A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.

Resonance NLS solitons as black holes in Madelung fluid

Envelope solitons of the Nonlinear Schrodinger equation (NLS) under quantum potentials influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s 1, to the reaction–diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1+1)-dimensional Jackiw–Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space–time metric. The space–time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.

Global solvability of the derivative nonlinear Schrödinger equation

The derivative nonlinear Schrodinger equation (DNLS) iqt = qxx ± (q*q2)x, <I = q(x,t), i = A77xr< ?*(z) _ Jffjt was first derived by plasma physicists (9, 10). This equation was used to in- terpret the propagation of circular polarized nonlinear Alfven waves in plasma. Kaup and Newell obtained the soliton solutions of DNLS in 1978 (5). The au- thor obtained the local solvability of DNLS in his dissertation (6). In this paper we obtain global existence (in time /) of Schwartz class solutions of DNLS if the Z.2-norm of the generic initial data q(x, 0) is bounded.

Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons

Abstract This paper is devoted to the semiclassical limit of the one-dimensional Schrodinger equation with current nonlinearity and Sobolev regularity, before shocks appear in the limit system. In this limit, the modified Euler equations are recovered. The strictly hyperbolicity and genuine nonlinearity are proved for the limit system wherever the Riemann invariants remain distinct. The dispersionless equation and its deformation which is the quantum potential perturbation of JNLS equation are also derived.

Self-Dual Vortices in Chern–Simons Hydrodynamics

AbstractThe classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie–Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant

Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions: Hirota bilinear method

\hbar

The behaviour of solutions of NLS equation of derivative type in the semiclassical limit

for the deformed strength