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 expfunction 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 Bäcklund transformation. MSC: 35Q58, 37K10, 35Q53

A new resonance version of NLS equation is found and embedded to the reactiondiffusion system, equivalent to the anti-de Sitter valued Heisenberg model, realizing a particular gauge fixing condition of the Jackiw-Teitelboim gravity. The space-time points where dispersion change the sign correspond to the event horizon, and the soliton solutions to the AdS black holes. The soliton with velocity bounded above describes evolution on the hyperboloid with nontrivial winding number and create under collisions the resonance states with a specific life time. Resonance NLS equation.The connection between black hole physics and the theory of supersonic acoustic flow was established by Unruh 1 and has been developed to investigate the Hawking radiation and other phenomena for understanding quantum gravity 2 . Recently, the similar idea for simulation of quantum effects related to event horizons and ergoregions but in superfluids, which in contrast to the usual liquids allow nondissipative motion of the flow, was proposed 3 . In this case a "superluminally" moving inhomogeneity of the order parameter like solitons, provides black holes like quasi-equilibrium states, exhibiting an event horizon. Although Jacobson and Volovik considered a simplified profile of soliton and mentioned unimportance of exact structure of the solution, an exactly soluble model of solitons related to black holes allows one to describe the scattering process of black holes and corresponding quantum phenomena. Very recently, some attempt was done to describe solitons of integrable models like the Liouville 4 , the Sine-Gordon 5,15 and the ReactionDiffusion system 6,7 , as black holes of Jackiw-Teitelboim (JT) gravity. In the last paper the scattering of two identical soliton-like structures called dissipatons, relating to the black holes and creating the metastable state was considered. Continuing in this direction, in the present paper we present a new integrable version of the Nonlinear Schrödinger equation (NLS) in 1+1 dimensions, admitting solitons with a rich resonance scattering phenomenology and interpreted as black holes of JT

It is shown that a generalized nonlinear Schrödinger equation proposed by Malomed and Stenflo admits, for a specific range of parameters, resonant soliton interaction. The equation is transformed to the 'resonant' nonlinear Schrödinger equation, as originally introduced to describe black holes in a Madelung fluid and recently derived in the context of uniaxial wave propagation in a cold collisionless plasma. A Hirota bilinear representation is obtained and soliton solutions are thereby derived. The one-soliton solution interpretation in terms of a black hole in two-dimensional spacetime is given. For the twosoliton solution, resonant interactions of several kinds are found. The addition of a quantum potential term is considered and the reduction is obtained to the resonant NLS equation.

Physically relevant soliton solutions of the resonant nonlinear Schrodinger (RNLS) equation with nontrivial boundary conditions, recently proposed for description of uniaxial waves in a cold collisionless plasma, are considered in the Hirota bilinear approach. By the Madelung representation, the model transformed to the reaction-diffusion analog of the NLS equation for which the bilinear representation, soliton solutions and their mutual interactions are studied.

This paper is devoted to the semiclassical limit of the one-dimensional Schr€ o odinger 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.