We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

We study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

This paper is concerned with the Cauchy problem for a nonlinear Schrödinger equation with a harmonic potential and exponential growth nonlinearity in two space dimensions. In the defocusing case, global well-posedness is obtained. In the focusing case, existence of nonglobal solutions is discussed via potential-well arguments.

The non-linear Schrödinger systems arise from many important physical branches. In this paper, employing the ‘I-method’, we prove the global-in-time well-posedness for a coupled non-linear Schrödinger system in Hs(n) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math Res. Lett. 9, 2002, 659–682) to the system.

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann–Hilbert problem whose solution gives the solution of our initial–boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift–Zhou methods.

This paper is concerned with the critical nonlinear Gross–Pitaevskii equation, which describes the attractive Bose–Einstein condensate under a magnetic trap. We derive a sharp threshold between the global existence and the blowing-up of the system. Furthermore, we answer the question: how small are the initial data, such that the system has global solutions for the nonlinear critical power $p=1+(4/N)$?