We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959 Sov. Phys. JETP8 282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in —strong in wavefunction and velocity, and weak in density.
ISSN: 1361-6544
Published jointly with the London Mathematical Society, Nonlinearity covers the interdisciplinary nature of nonlinear science, featuring topics which range from physics, mathematics and engineering through to biological sciences.
Cover credit: Dan J Hill et al 2023 36 2567.
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Juhi Jang et al 2024 Nonlinearity 37 065009
Thiago Carvalho Corso 2024 Nonlinearity 37 065003
In this article, we analyse the Dyson equation for the density–density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterise the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial DDRF and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation and the adiabatic local density approximation. In particular, these results show that neither of these approximations can shift the ionisation threshold of the Kohn–Sham system.
Jaemin Park 2024 Nonlinearity 37 065001
In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented example consists of two disjoint, simply connected patches. The rates of growth for both curvature and perimeter in this example are at least algebraic.
Hongwei Zhang et al 2024 Nonlinearity 37 065011
In this paper, we consider a class of wave-Hartree equations on a bounded smooth convex domain with Dirichlet boundary condition. We prove the local existence of solutions in the natural energy space by using the standard Galërkin method. The results on global existence and nonexistence of solutions are obtained mainly by means of the potential well theory and concavity method.
Albert Ai and Ovidiu-Neculai Avadanei 2024 Nonlinearity 37 055022
We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal.3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation's nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.
Humberto Ramos Quoirin et al 2024 Nonlinearity 37 065010
We look for critical points with prescribed energy for the family of even functionals , where are C1 functionals on a Banach space X, and . For a given and several classes of , we prove the existence of infinitely many couples such that More generally, we analyse the structure of the solution set of the problem with respect to µ and c. In particular, we show that the maps are continuous, which gives rise to a family of energy curves for this problem. The analysis of these curves provide us with several bifurcation and multiplicity type results, which are then applied to some elliptic problems. Our approach is based on the nonlinear generalized Rayleigh quotient method developed in Il'yasov (2017 Topol. Methods Nonlinear Anal.49 683–714).
Xuan Kien Phung 2024 Nonlinearity 37 065012
We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata (CA) over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform CA. A strengthened surjunctivity result is also obtained for multi-dimensional CA over strongly irreducible subshifts of finite type.
Sebastian Wieczorek et al 2023 Nonlinearity 36 3238
Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. This paper develops an accessible mathematical framework for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular R-tipping thresholds. These thresholds are anchored at infinity by regular R-tipping edge states: compact normally hyperbolic invariant sets of the autonomous future limit system that have one unstable direction, orientable stable manifold, and lie on a basin boundary. We define R-tipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the autonomous future limit system that is not an attractor. We focus on the case when the limit set is a regular edge state, introduce the concept of edge tails, and rigorously classify R-tipping into reversible, irreversible, and degenerate cases. The central idea is to use the autonomous dynamics of the future limit system to analyse R-tipping in the nonautonomous system. We compactify the original nonautonomous system to include the limiting autonomous dynamics. Considering regular R-tipping edge states that are equilibria allows us to prove two results. First, we give sufficient conditions for the occurrence of R-tipping in terms of easily testable properties of the frozen system and input variation. Second, we give necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system.
Tomáš Dohnal et al 2024 Nonlinearity 37 055005
Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.
Hua Zhang et al 2023 Nonlinearity 36 4585
The movements of species in a river are driven by random diffusion, unidirectional water flow, and cognitive judgement with spatial memory. In this paper, we formulate a reaction–diffusion–advection model with memory-based diffusion and homogeneous Dirichlet boundary conditions. The existence of a nonconstant positive steady state is proven. We obtain the linear stability of the steady state by analysing the eigenvalues of the associated linear operator: the nonconstant steady state can always be linearly stable regardless of the memory delay, while the model can also possess Hopf bifurcation as the memory delay varies. Moreover, theoretical and numerical results show that large advection annihilates oscillation patterns and drives the species to concentrate downstream.
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Rémi Carles 2024 Nonlinearity 37 065017
We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.
Yinbin Deng and Xian Yang 2024 Nonlinearity 37 065016
In the paper, we study the global higher regularity and decay estimates of the positive solutions for the following fractional equations where , , and is the fractional Laplacian. Let Q be a positive solution of (0.1). We prove that and obtain the decay estimates of DkQ as for all and . The argument relies on the Bessel kernel, comparison principle, Fourier analysis and iteration methods.
Ale Jan Homburg et al 2024 Nonlinearity 37 065015
Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in iterated function systems, and involves a superattracting periodic orbit. This paper will provide and study examples of iterated function systems by two rational maps on the Riemann sphere that give rise to critical intermittency. The main ingredient for this is a superattracting fixed point for one map that is mapped onto a common repelling fixed point by the other map. We include a study of topological properties such as topological transitivity.
Xiang Bai et al 2024 Nonlinearity 37 065014
In this paper, we study the asymptotic stability of the rarefaction wave for the one-dimensional compressible Euler system with nonlocal velocity alignment. Namely, for the initial data approaching to rarefaction wave, we prove the corresponding solution converges toward the rarefaction wave. Moreover, we obtain this system has weak alignment behavior. We develop some promoted estimates for the smooth approximate rarefaction wave and new a priori estimates by Fourier analysis tools. Moreover, we introduce the weighted energy method and Besov spaces to obtain the key high-order derivative estimates, in which we overcome the difficulties caused by the nonlocal velocity alignment. It is worth mentioning that this is the first stability result of rarefaction wave for compressible Euler system with velocity alignment.
Shintaro Suzuki 2024 Nonlinearity 37 065013
We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval (mod 1), , β > 0, which are the so-called beta-transformations. For such a random dynamical system, including the case that it is generated by uncountably many maps, we give an explicit formula for the density function of a unique stationary measure under the assumption that the random dynamics is expanding in mean. As an application, in the case that the random dynamics is generated by finitely many maps and the maps are chosen according to a Bernoulli measure, we show that the density function is analytic as a function of parameter in the Bernoulli measure and give its derivative explicitly. Furthermore, for a non-i.i.d. random dynamical system of beta-transformations, we also give an explicit formula for the random densities of a unique absolutely continuous invariant measure under a certain strong expanding condition or under the assumption that the maps randomly chosen are close to the beta-transformation for a non-simple number in the sense of parameter β.
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Ryan Goh and Arnd Scheel 2023 Nonlinearity 36 R1
Pattern forming systems allow for a wealth of states, where wavelengths and orientation of patterns varies and defects disrupt patches of monocrystalline regions. Growth of patterns has long been recognized as a strong selection mechanism. We present here recent and new results on the selection of patterns in situations where the pattern-forming region expands in time. The wealth of phenomena is roughly organised in bifurcation diagrams that depict wavenumbers of selected crystalline states as functions of growth rates. We show how a broad set of mathematical and numerical tools can help shed light into the complexity of this selection process.
Ali Tahzibi 2021 Nonlinearity 34 R75
In this survey we recall basic notions of disintegration of measures and entropy along unstable laminations. We review some roles of unstable entropy in smooth ergodic theory including the so-called invariance principle, Margulis construction of measures of maximal entropy, physical measures and rigidity. We also give some new examples and pose some open problems.
Thomas Bothner 2021 Nonlinearity 34 R1
This article is firstly a historic review of the theory of Riemann–Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert's 21st problem and Plemelj's work associated with it. The secondary purpose of this note is to invite a new generation of mathematicians to the fascinating world of Riemann–Hilbert techniques and their modern appearances in nonlinear mathematical physics. We set out to achieve this goal with six examples, including a new proof of the integro-differential Painlevé-II formula of Amir et al (2011 Commun. Pure Appl. Math. 64 466–537) that enters in the description of the Kardar–Parisi–Zhang crossover distribution. Parts of this text are based on the author's Szegő prize lecture at the 15th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) in Hagenberg, Austria.
Valerio Lucarini and Tamás Bódai 2020 Nonlinearity 33 R59
For a wide range of values of the intensity of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak-noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, which provides a possible alternative with respect to the edge-tracking algorithm.
D Lannes 2020 Nonlinearity 33 R1
We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the 'turbulent' and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre–Green–Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe–Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d = 1; among them are the KdV, BBM, Camassa–Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.
Open all abstracts, in this tab
Ale Jan Homburg et al 2024 Nonlinearity 37 065015
Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in iterated function systems, and involves a superattracting periodic orbit. This paper will provide and study examples of iterated function systems by two rational maps on the Riemann sphere that give rise to critical intermittency. The main ingredient for this is a superattracting fixed point for one map that is mapped onto a common repelling fixed point by the other map. We include a study of topological properties such as topological transitivity.
Xuan Kien Phung 2024 Nonlinearity 37 065012
We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata (CA) over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform CA. A strengthened surjunctivity result is also obtained for multi-dimensional CA over strongly irreducible subshifts of finite type.
Juhi Jang et al 2024 Nonlinearity 37 065009
We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959 Sov. Phys. JETP8 282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in —strong in wavefunction and velocity, and weak in density.
Thiago Carvalho Corso 2024 Nonlinearity 37 065003
In this article, we analyse the Dyson equation for the density–density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterise the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial DDRF and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation and the adiabatic local density approximation. In particular, these results show that neither of these approximations can shift the ionisation threshold of the Kohn–Sham system.
Jaemin Park 2024 Nonlinearity 37 065001
In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented example consists of two disjoint, simply connected patches. The rates of growth for both curvature and perimeter in this example are at least algebraic.
Albert Ai and Ovidiu-Neculai Avadanei 2024 Nonlinearity 37 055022
We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal.3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation's nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.
Tomáš Dohnal et al 2024 Nonlinearity 37 055005
Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.
N Chaudhuri et al 2024 Nonlinearity 37 045018
In this study, we analyse the famous Aw–Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called 'hard congestion limit', and show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is also illustrated numerically using a numerical scheme proposed for the model studied. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contacts between the grains.
Gabriele Bruell et al 2024 Nonlinearity 37 045016
We study stationary, periodic solutions to the thermocapillary thin-film model which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.
Lewis Napper et al 2024 Nonlinearity 37 045012
We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.