Numerical Methods for the Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation is one of the most successful models in mathematical physics. It governs the envelope of a weakly nonlinear dispersive wave in one spatial dimension, appearing in contexts as different as optical fibre communication, Bose–Einstein condensates, deep-water wave propagation, and plasma physics. Its one-dimensional defocusing cousin, the Gross–Pitaevskii equation, describes superfluid dynamics in a dilute atomic gas. What makes both equations remarkable is not their breadth of application but their depth of mathematical structure: they are exactly solvable by the inverse scattering transform, and their solutions — solitons, breathers, rogue waves, cnoidal waves — can be written down in closed form.

This book is about computing those solutions numerically, and computing them well. The approach throughout is the Method of Lines: the spatial variable is discretised by a pseudospectral operator (DST-I spline collocation for problems with Dirichlet boundary conditions, Fourier pseudospectral for problems with periodic ones), leaving an autonomous system of ordinary differential equations that is integrated in time by an explicit adaptive Runge–Kutta solver from SciPy. Every chapter applies this pipeline to a different exact solution, verifies the numerical output against the closed form to pointwise accuracy $10^{-10}$ or better, and monitors the conserved quantities — $L^2$ norm, momentum, and Hamiltonian energy — as independent diagnostics. The result, accumulated over fourteen chapters, is a complete computational atlas of the NLS/GPE solution zoo.

The book is organised in a single progression. Chapters 1 through 3 treat the elementary localised solutions of the focusing and defocusing NLS: the bright soliton, the black soliton, and the grey soliton family. Chapters 4 and 5 introduce the Darboux transformation and use it to construct multi-soliton exact solutions: the elastic two-soliton collision and the two-soliton bound state (soliton molecule). Chapters 6 and 7 leave the soliton family and treat the spatially periodic cnoidal waves via Jacobi elliptic functions, first with Dirichlet and then with periodic boundary conditions. Chapter 8 studies modulational instability — the Benjamin–Feir mechanism by which a plane wave amplifies sidebands exponentially — and introduces a different verification strategy based on linear growth rates rather than a pointwise exact solution. Chapters 9 through 11 treat the rogue-wave hierarchy sitting on a plane-wave background: the Akhmediev breather (periodic in space), the Kuznetsov–Ma breather (periodic in time), and the Peregrine soliton (localised in both variables), the latter being the degenerate limit of the former two. Chapter 12 extends the pipeline to the Manakov vector NLS, showing how the two-component system inherits elastic collisions and exact solutions from the scalar theory. Chapter 13 constructs the three-soliton collision via a three-step Darboux transform and measures the pairwise phase shifts analytically. Chapter 14 closes Part I with the second-order rational rogue wave, the $N = 2$ member of the Peregrine hierarchy, whose peak amplitude is five times the background. Chapter 15 is an epilogue: it explains the mathematical reasons the pipeline works, identifies its honest limits, and sketches ten natural extensions that go beyond the exact-solution regime.