Construction of multi-bubble solutions for the critical gKdV equation

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that u(t) - \sum_{k=1}^K \frac {\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{\lambda_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.Comment: 70 page

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

Blow-up dynamics for smooth finite energy radial data solutions to the self-dual Chern-Simons-Schr\"odinger equation

We consider the finite-time blow-up dynamics of solutions to the self-dual Chern-Simons-Schr\"odinger (CSS) equation (also referred to as the Jackiw-Pi model) near the radial soliton $Q$ with the least $L^{2}$-norm (ground state). While a formal application of pseudoconformal symmetry to $Q$ gives rise to an $L^{2}$-continuous curve of initial data sets whose solutions blow up in finite time, they all have infinite energy due to the slow spatial decay of $Q$. In this paper, we exhibit initial data sets that are smooth finite energy radial perturbations of $Q$, whose solutions blow up in finite time. Interestingly, their blow-up rate differs from the pseudoconformal rate by a power of logarithm. Applying pseudoconformal symmetry in reverse, this also yields a first example of an infinite-time blow-up solution, whose blow-up profile contracts at a logarithmic rate. Our analysis builds upon the ideas of previous works of the first two authors on (CSS) [21,22], as well as the celebrated works on energy-critical geometric equations by Merle, Rapha\"el, and Rodnianski [33,38]. A notable feature of this paper is a systematic use of nonlinear covariant conjugations by the covariant Cauchy-Riemann operators in all parts of the argument. This not only overcomes the nonlocality of the problem, which is the principal challenge for (CSS), but also simplifies the structure of nonlinearity arising in the proof.Comment: 80 page

Infinite hierarchy of nonlinear Schrödinger equations and their solutions

We study the infinite integrable nonlinear Schrödinger equation (NLSE) hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, AB breathers, Kuznetsov-Ma breathers, periodic solutions and rogue wave solutions for this infinite order hierarchy. We find that 'even' order equations in the set affect phase and 'stretching factors' in the solutions, while 'odd' order equations affect the velocities. Hence 'odd' order equation solutions can be real functions, while 'even' order equation solutions are always complex

Bifurcation analysis and exact solutions for a class of generalized time-space fractional nonlinear Schrödinger equations

In this work, we focus on a class of generalized time-space fractional nonlinear Schrödinger equations arising in mathematical physics. After utilizing the general mapping deformation method and theory of planar dynamical systems with the aid of symbolic computation, abundant new exact complex doubly periodic solutions, solitary wave solutions and rational function solutions are obtained. Some of them are found for the first time and can be degenerated to trigonometric function solutions. Furthermore, by applying the bifurcation theory method, the periodic wave solutions and traveling wave solutions with the corresponding phase orbits are easily obtained. Moreover, some numerical simulations of these solutions are portrayed, showing the novelty and visibility of the dynamical structure and propagation behavior of this model

Energy conversion efficiency from a high order soliton to fundamental solitons in presence of Raman scattering

We formulate the energy conversion efficiency from a high-order soliton to fundamental solitons by including the influence of interpulse Raman scattering in the fission process. The proposed analytical formula agrees closely with numerical results of the generalized nonlinear Schrodinger equation as well as to experimental results, while the resulting formulation significantly alters the energy conversion efficiency predicted by the Raman-independent inverse scattering method. We also calculate the energy conversion efficiency in materials of different Raman gain profiles such as silica, ZBLAN and chalcogenide glasses (As2S3 and As2Se3). It is predicted that ZBLAN glass leads to the largest energy conversion efficiency of all four materials. The energy conversion efficiency is a notion of utmost practical interest for the design of wavelength converters and supercontinuum generation systems based on the dynamics of soliton self-frequency shift.Comment: To be published in JOSA

Omega network hash construction

Cryptographic hash functions are very common and important cryptographic primitives. They are commonly used for data integrity checking and data authentication. Their architecture is based on the Merkle-Damgard construction, which takes in a variablelength input and produces a fixed-length hash value. The basic Merkle-Damgard construction runs over the input sequentially, which can lead to problems when the input size is large since the computation time increases linearly. Therefore, an alternative architecture which can reduce the computation time is needed, especially in today's world where multi-core processors and multithreaded programming are common. An Omega Network Hash Construction that can run parallel on multi-core machine has been proposed as alternative hash function's construction. The Omega Network Hash Construction performs better than the Merkle-Damgard construction, and its permutation architecture shows that its security level in term of producing randomness digest value is better than Merkle-Damgard construction

Experiments on synthetic dimensions in photonics

The first and introductory section of the dissertation presents the working principle of a one- and two-dimensional photonic mesh lattice based on the time-multiplexing technique. The basis of a random walk interrelated to the corresponding light and quantum walk is comprehensively discussed as well. The second part of the dissertation consists of three experiments on a one-dimensional photonic mesh lattice. Firstly, the Kapitza-based guiding light project models the Kapitza potential as a continuous Pauli-Schrödinger-like equation and presents an experimental observation of light localization when the transverse modulation is bell-shaped but with a vanishing average along the propagation direction. Secondly, the optical thermodynamics project experimentally demonstrates for the first time that any given initial modal occupancy reaches thermal equilibrium by following a Rayleigh-Jeans distribution when propagates through a multimodal photonic mesh lattice with weak nonlinearity. Remarkably, the final modal occupancy possesses a unique temperature and chemical potential that have nothing to do with the actual thermal environment. Finally, the quantum interference project discusses an experimental all-optical architecture based on a coupled-fiber loop for generating and processing time-bin entangled single-photon pairs. Besides, it shows coincidence-to-accidental ratio and quantum interference measurements relying on the phase modulation of those time bins. The third part of the dissertation comprises two experiments on a two-dimensional photonic mesh lattice. The first project discusses the experimental realization of a two-dimensional mesh lattice employing short- and long-range interaction. To some extent, the second project presents a nonconservative system based on a two-dimensional photonic mesh lattice exploiting parity-time (PT) symmetry