We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude $q_o \gt 0$ as $x\to \pm \infty$ by implementing numerically the corresponding inverse scattering transform (IST). The Riemann–Hilbert problem associated with the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift–Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the $(x,t)$-plane depending on whether $|x/(2t)| \lt q_o$ or $|x/(2t)| \gt q_o$, as $t \to \infty$.

The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $\beta \gt 1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric ‘fully localized’ or ‘lump’ solitary waves which decay to zero in all spatial directions. The full-dispersion KP-I equation is obtained by retaining the exact dispersion relation in the modelling from the water-wave problem. In this paper, we show that the FDKP-I equation also has a family of symmetric fully localized solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.

We determine spatially-bounded rogue waves in the Davey–Stewartson I equation. We show that these rogue waves can be obtained when a single or multiple internal parameters in the higher-order rational solution of the Davey–Stewartson I equation are real and large, and the order-index vector of this higher-order rational solution has even length and comprises pairs of the form $(2n, 2n+1)$, where $n$ is a positive integer. Under these conditions and another nondegeneracy condition on the root curve of a certain double-real-variable polynomial, the higher-order rational solution will exhibit spatially-bounded rogue waves that arise from a uniform background with some time-varying lumps on it, reach high amplitude in limited space, and then disappear into the same background again. The crests of these rogue waves form a single or multiple closed curves that are generically disconnected from each other on the spatial plane, and are analytically predicted by the root curve mentioned above. We also derive uniformly-valid asymptotic approximations for these spatially-bounded rogue waves in the large-parameter regime. Near the crests of these rogue waves, these asymptotic approximations reduce to simple expressions. Our asymptotic approximations of these rogue waves are compared to true solutions, and good agreement is demonstrated.

We report the experimental measurement of the density of states (DOS) associated with the soliton gas emerging during the development of the noise-induced modulation instability (MI) in optical fibres. By employing a time-lens-based heterodyne detection technique (SEAHORSE), we reconstruct the complex optical field and compute its nonlinear discrete spectrum within the framework of the inverse scattering transform. Our results show that, at early stages of the MI, the DOS matches the Weyl distribution predicted for an ‘ideal’ critically dense soliton gas, thereby confirming the relevance of the SG description for this nonlinear random wave regime. At larger effective propagation distances, we observe a progressive deformation of the DOS in the complex plane. We compare these observations with numerical simulations of a generalised nonlinear Schrödinger equation that includes losses, third-order dispersion and stimulated Raman scattering (SRS). Our simulations reproduce the main experimental trends and demonstrate that SRS is the dominant mechanism responsible for the spectral deformation. These findings highlight the need to extend the kinetic theory of soliton gas beyond purely integrable evolutions. In particular, our results call for a generalised kinetic equation (or generalised hydrodynamics description) that accounts for weak non-integrable perturbations such as SRS.

We investigate the long-time behaviour of solutions with small initial data to the viscoelastic Klein–Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. A key difficulty arises from the viscous damping, which introduces a nonzero real part in the coefficient of the remaining critical resonant term. By carefully decomposing the solution into a Gaussian profile and a zero-mean remainder, we isolate the leading-order resonant dynamics. We identify a sign condition under which these dynamics are of absorption type. In this case, we prove global-in-time existence and enhanced diffusive decay for solutions with small initial data, and we characterize their leading-order asymptotics. Conversely, when this condition fails, the reduced resonant dynamics exhibit finite-time blow-up. In this regime, global existence is not expected, but solutions with small initial data persist and decay diffusively over exponentially long time intervals.

While real-valued solutions of the Korteweg–de Vries (KdV) equation have been studied extensively over the past 50 years, much less attention has been devoted to solution behaviour in the complex plane. Here we consider the analytic continuation of real solutions of KdV and investigate the role that complex-plane singularities play in early time solutions on the real line. We apply techniques of exponential asymptotics to derive the small-time behaviour for dispersive waves that propagate in one direction, and demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of double-pole singularities of the initial condition in the complex plane. Using matched asymptotic expansions in the limit $t\rightarrow 0^+$, we show how complex singularities of the time-dependent solution of the KdV equation emerge from these double-pole singularities. Generically, their speed as they move from their initial position is of $\mathcal{O}(t^{-2/3})$, while the direction in which these singularities propagate initially is dictated by a Painlevé II (P$_{\mathrm{II}}$) problem with decreasing tritronquée solutions. The well-known $N$-soliton solutions of KdV correspond to rational solutions of P$_{\mathrm{II}}$ with a finite number of singularities; otherwise, we postulate that infinitely many complex-plane singularities of KdV solutions are born at each double-pole singularity of the initial condition. We also provide asymptotic results for some non-generic cases in which singularities propagate more slowly than in the generic case. Our study makes progress towards the goal of providing a complete description of KdV solutions in the complex plane and, in turn, of relating this behaviour to the solution on the real line.

Periodic water waves of permanent form travelling at constant speed, the so-called Stokes waves, are studied in water of fixed finite depth using methods previously used in water of infinite depth. We apply our methods to waves of varying steepness over a range of fixed depths in order to determine how a number of physical quantities related to the waves change as the steepness of the waves increases. Finally, we examine the complex singularities outside of their domain of definition when the waves are considered as a function of a conformal variable.

We study the continuum limit of the motion of agents in the plane driven by competing short-range repulsion and long-range attractive forces. At a critical parameter value, we find destabilization of a trivial branch of uniformly distributed solutions and analyse bifurcating solutions. Curiously, the bifurcating branch is vertical, leading to a reversible, non-hysteretic phase transition. Near the bifurcation point, we demonstrate scaling laws for the size of vacuum regions, which can form fissures or bubbles. We also study the effect of small noise and the eventual topological transition from vacuum bubbles to isolated particle clusters.

The goal of this work is to revisit the eigenfunction-expansion-based perturbation theory of the defocusing nonlinear Schrödinger equation on a nonzero background, and develop it to correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the soliton sides. Proof of the closure of the squared eigenfunctions is provided, and the complete set of eigenfunctions of the linearisation operator is used to expand the first-order perturbation solution. Our closure/completeness relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions. Using the one-soliton closure relation and its correct discrete eigenmodes, the slow-time evolution equations of the soliton parameters are determined. Moreover, the first-order correction integral to the dark soliton is shown to contain a pole due to singularities of the scattering data at the branch points. Analysis of this integral leads to predictions for the shelves, as well as a formula for the slow-time evolution of the soliton’s phase, which in turn allows one to determine the slow-time dependence of the soliton centre. All the results are corroborated by direct numerical simulations and compared with earlier results.

In the present work, we examine the dynamics of a model for oscillons in one-dimensional space-time field theories with a cubic nonlinearity. We utilize a reduction of the model to first and third harmonics, which leads to a reduced partial differential equation (PDE) system whose steady states are candidates for the original PDE oscillons. We analyse the steady states of this model and their stability, via tools such as index theory. We develop suitable functionals needed for the study of such stationary states, as well as an analogue of the famous Vakhitov–Kolokolov criterion for a quantity whose change of monotonicity reflects a change of stability. Then, we test the relevant predictions, over the full range of oscillon frequencies, through systematic numerical computations of both the reduced model, its steady states and stability, and also of the original PDE model, identifying its time-periodic oscillon solution. Our results yield some significant connections with previous studies, but also some key new insights, both on the reduced system and the dynamics of the original system.

We examine a class of reaction-diffusion systems where saturation drives pattern formation. A prime example is the classical model of mussel–algae interactions. We asymptotically construct a pattern consisting of $N$ spikes. Unlike many other models where the inner spike profile is described by a nonlinear ODE $w^{\prime\prime}-w+w^{p}=0$, here it is described by a linear ODE $w^{\prime\prime}+w-1=0.$ We then study the stability of the resulting patterns. A new non-local eigenvalue problem is derived which controls the structural stability of $N$ such spikes. We rigorously prove its stability in the relevant regime, taking advantage of the explicit formula for the shape of the spike. Self-replication is also observed, and we derive a new ‘core problem’ which captures this phenomenon. Finally, the asymptotic stability band for the existence of $N$ spikes is given. Our methods are applicable to a wide range of systems that exhibit saturation; we provide a variant of the Schnakenberg model as another example. Numerical simulations confirm our asymptotic predictions.

We prove the existence of solitary waves in a lattice where all particles interact with each other by pairwise repulsive forces that decay with distance. The variational existence proof is based on constrained optimization and provides a one-parameter family of unimodal solutions. We also describe the asymptotic behaviour of large, fast, high-energy waves.

We investigate the dynamical role of internal vibrational modes in the self-dual Abelian-Higgs model, focusing on how Derrick-type breathing excitations modify vortex dynamics and scattering processes. While unexcited BPS vortices follow geodesic motion leading to the well-known $90^\circ$ scattering, initially excited vortices experience vibration-induced forces arising from the spectral flow of internal modes as the inter-vortex distance changes. This mechanism intensifies the resonant energy transfer between translational and vibrational channels, producing non-adiabatic dynamics, superelastic collisions and fractal structures in scattering diagrams.

A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi–Pasta–Ulam lattice with a general cubic interaction force. Explicit periodic travelling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic travelling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic velocities is shown to lead to modulational instability of periodic travelling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic travelling waves and initial value problems that also reveal additional short-wavelength instabilities.

With possible exceptions in quantum mechanics, measurable physical quantities do not involve complex numbers. However, most standard and special functions in the applied sciences are analytic functions. Focusing on exact mathematical techniques, a famous quote notes that ‘The shortest path between two truths in the real domain passes through the complex domain’. It therefore becomes natural to ask if this shortest path can benefit from finite difference formulas and numerical quadrature methods that are specifically designed for the complex plane. Several such opportunities have recently been found. While the concepts of complex variables and of analyticity do not readily generalize beyond 2-D, the new numerical methods that are described here can be adapted to harmonic functions in higher dimensions.

We study the modulational stability problem for the travelling periodic waves (called Stokes waves) in an infinitely deep fluid by using pseudo-differential operators in conformal variables. We derive the criteria and the normal forms for four bifurcations which are repeated recurrently when the steepness of the Stokes wave is increased towards the highest wave with the peaked profile. The four bifurcations are observed in the following order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration of figure-8 bands resulting in vertical slopes, (c) new circular bands around the origin appearing at each period-doubling bifurcation and (d) reconnection of figure-$\infty$ bands at each extremal point of energy. Our work uses the analytic theory of Stokes waves developed previously for Babenko’s equation. The novelty of our work is the analytic extension of the modulational stability problem for singular pseudo-differential operators in terms of the Floquet parameter. The derivation of the normal form uses some structural assumptions which are known to be true for the Stokes waves. For the first and second bifurcation cycles, we compute numerically with a higher-order accuracy the actual values of wave steepness for which the structural assumptions are satisfied and the numerical coefficients of the normal forms to show the excellent agreement between the normal form theory and the numerical approximations of the spectral bands.

The modulational instability of the focusing nonlinear Schrödinger equation has been proposed as a possible mechanism in the formation of rogue waves on the ocean and other nonlinear dispersive systems. In this paper, the focus is on the pole dynamics associated with the modulational instability. A range of pole behaviours is identified, ranging from simple periodic motion along the imaginary axis when the nonlinear parameter is relatively small, to motions that trace complicated trajectories in the complex plane when the parameter is larger. Possibly of relevance to rogue wave theories is the situation where several poles have a near-collision close to the real axis, leading to a disproportionately high peak in the modulus of the solution. In the absence of explicit solution formulas, this study is based on computational pole tracking using Fourier–Padé methods.

The nonstationary Schrödinger equation (NSE) with a special class of rational potentials is investigated via binary Darboux transformation. It is shown that in this case, the NSE corresponds to the focusing nonlinear Schrödinger equation (NLS): $ir_y=r_{xx}+|r|^2r$ where $r(x,y)$ is rational with non-vanishing boundary conditions: $|r| \to b$ as $|x| \to \infty$. These rational solutions of NLS are the so-called rogue wave solutions that have been studied extensively for the last 15 years or so. Furthermore, these rational potentials of the NSE are also shown to correspond to a special family of rational, multi-lump solutions of the Kadomtsev–Petviasvili I equation with multiple peaks in the $xy$-plane and vanishing as $x^2+y^2 \to \infty$.

We study, numerically, the self-similar propagation of optical pulses – similaritons – governed by the non-linear Schrödinger equation with fractional-order dispersion and constant gain. Whereas in the classical, integer-order limit a Gaussian input relaxes to well-known parabolic similariton, we show that fractional dispersion, controlled by the Lévy index, fundamentally reshapes this evolution. As the Lévy index is reduced below its classical value, a standard Gaussian input evolves into parabolic, flat-top or two-peaked profiles, the last of which is a genuine consequence of fractionality and is absent in the integer-order case. We trace the physical origin of this two-peaked, hollowed core to the singular low-wavenumber behaviour of the fractional operator, and we demonstrate that it is suppressed in two distinct ways: by increasing the gain, which drives the pulse onto the parabolic attractor, and by broadening the initial condition, which reduces the reshaping that the pulse must undergo. A sufficiently stretched Gaussian input restores clean parabolic self-similar propagation across the entire range of fractional orders considered. Finally, we show that fractional similaritons develop pronounced, algebraically decaying heavy tails – in stark contrast to the exponential localization of the classical case – whose power-law exponent is fixed directly by the Lévy index, providing a clear signature of the underlying Lévy-type transport.