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Yair Zarmi

Publications and source records attributed to Yair Zarmi.

At least 19 recordsLinked to original sources
arXiv2020

Enhanced Algal Photosynthetic Photon Efficiency by Pulsed Light

We present experimental results demonstrating that, relative to continuous illumination, an increase of a factor of 3-10 in the photon efficiency of algal photo-synthesis is attainable via the judicious application of pulsed light for light intensities of practical interest (e.g., average-to-peak solar photon flux). We also propose a simple model that can account for all the measurements. The model (1) reflects the essential rate-limiting elements in bio-productivity, (2) incorporates the impact of photon arrival-time statistics and (3) accounts for how the enhancement in photon efficiency depends on the timescales of light pulsing and photon flux density. The key is avoiding clogging of the photosynthetic pathway by properly timing the light-dark cycles experienced by algal cells. We show how this can be realized with pulsed light sources, or by producing pulsed-light effects from continuous illumination via turbulent mixing in dense algal cultures in thin photo-bioreactors.

q-bio.QM↗
arXiv2018

Soliton-Generating $τ$-Functions Revisited

Within the framework of the Inverse-Scattering formalism and the Hirota algorithm, soliton solutions of evolution equations are images of τ-functions. Typically, the latter are expressed in terms of exponentials, the arguments of which are linear in the coordinates. Consequently, often, τ-functions are unbounded in space and time. However, they are not unique. Exploitation of their non-uniqueness uncovers physically interesting possibilities: 1) One can construct equivalent τ-functions, which generate the same traditional (Inverse-Scattering/Hirota)) soliton solutions, yet allow for the extension of the family of soliton solutions to a wider, parametric family, in which the traditional solutions are a subset. The parameters are shifts in individual soliton trajectories. 2) When two wave numbers in a multi-soliton solution are made to coincide, the reduction of the solution to one with a lower number of solitons is qualitatively different for solutions that are within the traditional subset and those that are outside this subset. 3) One can construct τ-functions that are bounded in space and time, in terms of which soliton solutions become images of localized sources.

nlin.SI↗
arXiv2017

Traveling waves and localized structures: An alternative view of nonlinear evolution equations

Given a nonlinear evolution equation in (1+n) dimensions, which has spatially extended traveling wave solutions, it can be extended into a system of two coupled equations, one of which generates the original traveling waves, and the other generates structures that are localized in the vicinity of the intersections of the traveling waves. This is achieved thanks to the observation that, as a direct consequence of the original evolution equation, a functional of its solution exists, which vanishes identically on the single-wave solution. This functional maps any multi-wave solution onto a structure that is confined to the vicinity of wave intersections. In the case of solitons in (1+1) dimensions, the structure is a collection of humps localized in the vicinity of soliton intersections. In higher space dimensions these structures move in space. For example, a two-front system in (1+3) dimensions is mapped onto an infinitely long and laterally bounded rod, which moves in a direction perpendicular to its longitudinal axis. The coupled systems corresponding to several known evolution equations in (1+1), (1+2) and (1+3) dimensions are reviewed.

nlin.SI↗
arXiv2014

Nonlinear quantum-mechanical system associated with Sine-Gordon equation in (1+2) dimensions

Despite the fact that it is not integrable, the 1 + 2-dimensional Sine-Gordon equation has N-soliton solutions, whose velocities are lower than the speed of light (c = 1), for all N greater than or equal to 1. Based on these solutions, a quantum-mechanical system is constructed over a Fock space of particles. The coordinate of each particle is an angle around the unit circle. U, a nonlinear functional of the particle number-operators, which obeys the Sine-Gordon equation in 1+2 dimensions, is construct-ed. Its eigenvalues on N-particle states in the Fock space are the slower-than-light, N-soliton solutions of the equation. A projection operator (a nonlinear functional of U), which vanishes on the single-particle subspace, is a mass-density generator. Its eigenvalues on multi-particle states play the role of the mass density of structures that emulate free, spatially extended, relativistic particles. The simplicity of the quantum-mechanical system allows for the incorporation of perturbations with particle interactions, which have the capacity to annihilate and create solitons - an effect that does not have a classical analog.

nlin.SI↗
arXiv2014

Spatially extended relativistic particles associated with multi-soliton solutions of the Sine-Gordon equation in more than one space dimension

Contrary to the decades-old understanding, SGn, the Sine-Gordon equation in (1+n) dimensions, has N-soliton solutions for any N >= 1, not only for n = 1, but also for n = 2 and 3. While SG1 solitons are confined to a line, SG2- and SG3-solitons are confined to a plane. An SG2-soliton solution moves rigidly with a constant velocity in the plane, and an SG3-solution moves rigidly with a constant velocity in the plane, and along the normal to the plane. A conservation law for the current density, obeyed by the single-SGn-soliton solution, is violated by all multi-soliton solutions. The violation manifests itself by generating vertices, structures that are localized around the soliton collision regions and decay exponentially in all directions in the (1+n)-dimensional space. In (1+1) dimensions, vertices evolve and then decay. In (1+2) and (1+3) dimensions, they move with the whole solution at its constant velocity, preserving their profiles, thereby emulating free, spatially extended, relativistic particles and bound states of such particles.

nlin.SI↗
arXiv2014

Sine-Gordon equation in higher dimensions: A fresh look at integrability

The Sine-Gordon equation is integrable in (1+1)-dimensional Minkowski and in 2-dimensional Euclidean spaces. In each case, it has a Lax pair, and a Hirota algorithm generates its N soliton solutions for all N greater than or equal to 1. The (1+2)-dimensional equation does not pass known integrability tests and does not have a Lax pair. Still, the Hirota algorithm generates N soliton solutions of that equation for all N greater than or equal to 1. Each multi-soliton solution propagates rigidly at a constant velocity, v. The solutions are divided into two unconnected subspaces: Solutions with v greater than or equal to c =1, and v smaller than c. Each subspace is connected by an invertible transformation (rotation plus dilation) to the space of soliton solutions of an integrable Sine-Gordon equation in two dimensions. The faster-than-light solutions are connected to the solutions in (1+1)-dimensional Minkowski space. The slower-than-light solutions are connected to the solutions in 2-dimensional Euclidean space by this transformation and also by Lorentz transformations. The Sine-Gordon equation in (1+3)-dimensional Minkowski space has a richer variety of solutions. Its slower-than-light solutions are connected to the solutions of the integrable equation in 2-dimensional Euclidean space. However, only a subset of its faster-than-light solutions is connected to the solutions of the integrable equation in (1+1)-dimensional Minkowski space.

nlin.SI↗
arXiv2014

On multi soliton solutions of the Sine-Gordon equation in more than one space dimension

The (1+1)-dimensional Sine-Gordon equation passes integrability tests commonly applied to nonlinear evolution equations. Its soliton solutions are obtained by a Hirota algorithm. In higher space-dimensions, the equation does not pass these tests. In this paper, using no more than the relativistic kinematics of the tachyonic momentum vectors, from which the soliton solutions are constructed through the Hirota algorithm, the existence and classification of N-soliton solutions of the (1+2)- and (1+3)-dimensional equations for all N greater than or equal to 1 are presented. In (1+2) dimensions, each multisoliton solution propagates rigidly at one velocity. The solutions are divided into two subsets: Solutions whose velocities are lower than the speed of light (c = 1), or are greater than or equal to c. In (1+3)-dimensions, multisoliton solutions are characterized by spatial structure and velocity composition. The spatial structure is either planar (rotated (1+2)-dimensional solutions), or genuinely three-dimensional - branes. Some solutions, planar or branes, propagate rigidly at one velocity, which is lower than, equal to, or higher than c. A subset of the branes contains hybrids, in which different clusters of solitons propagate at different velocities. Some velocities may be lower than c but some must be equal to, or greater than c. Finally, the speed of light cannot be approached from within the subset of slower-than-light solutions in both (1+2) and (1+3) dimensions.

nlin.SI↗
arXiv2014

Relativistic particle-like structures associated with multi-soliton solutions of (1+2)-dimensional Sine-Gordon equation

The Sine-Gordon equation in (1+2) dimensions has N-soliton solutions that propagate at velocities that are lower than the speed of light (c = 1), for any N greater tha or equal to 1. A first integral of the equation, which vanishes identically on the single soliton solution, maps multisoliton solutions onto structures that are localized around soliton junctions. The profile of such a structure obeys the (1+2)-dimensional linear wave equation, driven by a source term, which is constructed from a multisoliton solution of the Sine-Gordon equation. If the localized solutions of the source-driven wave equation are interpreted as mass densities, they emulate free, spatially extended, massive relativistic particles. This physical picture is summarized in terms of a Lagrangian density for a dynamical system, in which the Sine-Gordon equation and the linear wave equation are coupled by a small coupling term. The Euler-Lagrange equations of motion allow for solutions, which, in lowest order in the coupling constant are the soliton solutions of the Sine-Gordon equation, and the first-order component are the structures that emulate spatially extended relativistic particles.

nlin.SI↗
arXiv2013

Spatially extended particles hidden in line-soliton dynamics in more than one space dimension

A dynamics of spatially extended particles, hidden in the dynamics of line solitons in more than one space dimension, is revealed through conservation laws obeyed by the single-soliton solution. These are functions of the solution of a nonlinear evolution equation and its derivatives, which vanish for a single-soliton solution. They map multi-line-soliton solutions into systems of vertices - spatially extended structures, localized around the soliton-collision regions. In more than one space dimension, vertices move in space, emulating the dynamics of spatially extended particles. Examples are provided through the analysis of several soliton solutions of the Kadomtsev-Petviashvili (KP) equation in (1+2)-dimensions. The solution with one collision region is mapped onto a single vertex, which moves at a constant velocity, preserving its spatial structure, thereby emulating a free particle. In solutions with several collision regions, each region is mapped onto a vertex. When vertices are well separated, they also emulate free particles. They move in space, coalesce upon collision, and then split up. The total particle linear momentum is conserved through the collision in all solutions studied. However, depending on the soliton solution, particle masses and the total kinetic energy may change.

nlin.SI↗
arXiv2013

Vertex dynamics in multi-soliton solutions of Kadomtsev-Petviashvili II equation

A functional of the solution of the Kadomtsev-Petviashvili II equation maps multi-soliton solutions onto systems of vertices - structures that are localized around soliton junctions. A solution with one junction is mapped onto a single vertex, which emulates a free, spatially extended, particle. In solutions with several junctions, each junction is mapped onto a vertex. Moving in the x-y plane, the vertices collide, coalesce upon collision, and then split up. When well separated, they emulate free particles. Multi-soliton solutions, whose structure does not change under space-time inversion as for infinite times, are mapped onto vertex systems that undergo elastic collisions. Solutions, whose structure does change, are mapped onto systems that undergo inelastic collisions. The inelastic vertex collisions generated from the infinite family of (M,1) solutions (M external solitons, (M-2) Y-shaped soliton junctions, M greater than 3) play a unique role: The only definition of vertex mass consistent with momentum conservation in these collisions is the spatial integral of the vertex profile. This definition ensures, in addition, that, in these collisions, the total mass and kinetic energy due to the motion in the y-direction are conserved. In general, the kinetic energy due to the motion in the x-direction is not conserved in these collisions.

nlin.SI↗
arXiv2012

Static solitons, Lorentz invariance, and a new perspective on the integrability of the Sine Gordon equation in (1+2) dimensions

Contrary to the common understanding, the Sine-Gordon equation in (1+2) dimensions does have N-soliton solutions for any N. The Hirota algorithm allows for the construction of static N-soliton solutions (i.e., solutions that do not depend on time) of that equation for any N. Lorentz transforming the static solutions yields N-soliton solutions in any moving frame. They are scalar functions under Lorentz transformations. In an N-soliton solution in a moving frame, (N-2) of the (1+2)-dimensional momentum vectors of the solitons are linear combinations of the two remaining vectors.

nlin.SI↗
arXiv2012

A nonlinear quantum dynamical system of spin 1/2 particles based on the classical Sine-Gordon Equation

The Hirota transformation for the soliton solutions of the classical Sine-Gordon equation is suggestive of an extremely simple way for the construction of a nonlinear quantum-dynamical system of spin 1/2 particles that is equivalent to the classical system over the soliton sector. The soliton solution of the classical equation is mapped onto an operator, U, a nonlinear functional of the particle-number operators, that solves the classical equation. Multi-particle states in the Fock space are the eigenstates of U; the eigenvalues are the soliton solutions of the Sine-Gordon equation. The fact that solitons can have positive as well negative velocities is reflected by the characterization of particles in the Fock space by two quantum numbers: a wave number k, and a spin projection, σ (= +-1). Thanks to the simplicity of the construction, incorporation of particle interactions, which induce soliton effects that do not have a classical analog, is simple.

nlin.SI↗
arXiv2011

Quantized representation for Kadomtsev-Petviashvili equation on the soliton sector

Exploiting the known structure of soliton solutions, obtained through the Hirota transformation, a quantized representation of the Kadomtsev-Petviashvili (KP) equation on the soliton sector is constructed over a Fock space of particles, which may be either bosons or fermions. The classical solution is mapped into an operator, which also obeys the KP equation. The operator is constructed in terms of the particle-number operators. Classical soliton solutions are the expectation values of this operator in multi-particle states in the Fock space. The operator is equation-specific and the state in the Fock space is in one-to-one correspondence with the particular soliton solution.

nlin.SI↗
arXiv2010

Special polynomials and soliton dynamics

Special polynomials play a role in several aspects of soliton dynamics. These are differential polynomials in u, the solution of a nonlinear evolution equation, which vanish identically when u represents a single soliton. Local special polynomials contain only powers of u and its spatial derivatives. Non-local special polynomials contain, in addition, non-local entities (e.g., δx-1u). When u is a multiple-solitons solution, local special polynomials are localized in the vicinity of the soliton-collision region and fall off exponentially in all directions away from this region. Non-local ones are localized along soliton trajectories. Examples are presented of how, with the aid of local special polynomials, one can modify equations that have only a single-soliton solution into ones, which have that solution as well as, at least, a two-solitons solutions. Given an integrable equation, with the aid of local special polynomials, it is possible to find all evolution equations in higher scaling weights, which share the same single-soliton solution and are either integrable, or, at least, have a two-solitons solution. This is demonstrated for one or two consecutive scaling weights for a number of known equations. In the study of perturbed integrable equations, local special polynomials are responsible for inelastic soliton interactions generated by the perturbation in the multiple-soliton case, and for the (possible) loss of asymptotic integrability. Non-local special polynomials describe higher-order corrections to the solution, which are of an inelastic nature.

nlin.SI↗
arXiv2010

Quantized representation of some nonlinear integrable evolution equations on the soliton sector

The Hirota algorithm for solving several integrable nonlinear evolution equations is suggestive of a simple quantized representation of these equations and their soliton solutions over a Fock space of bosons or of fermions. The classical nonlinear wave equation becomes a nonlinear equation for an operator. The solution of this equation is constructed through the operator analog of the Hirota transformation. The classical N-solitons solution is the expectation value of the solution operator in an N-particle state in the Fock space.

nlin.SI↗
arXiv2008

Special features of the KdV-Sawada-Kotera equation

The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is developed through third order within the framework of a Normal Form analysis. It is shown that the equation is asymptotically integrable through the order considered. Focusing on the soliton sector, it is shown that the higher-order corrections in the Normal Form expansion represent purely inelastic KdV-soliton-collision processes, and vanish identically in the single-soliton limit. These characteristics are satisfied by the exact two-soliton solution of the KdV-Sawada-Kotera equation: The deviation of this solution from its KdV-type two-soliton approximation describes a purely inelastic scattering process: The incoming state is the faster KdV soliton. It propagates until it hits a localized perturbation, which causes its transformation into the outgoing state, the slower soliton. In addition, the effect of the perturbation on the exact two-soliton solution vanishes identically in the single-soliton limit (equal wave numbers for the two solitons).

nlin.SI↗
arXiv2008

Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of Plasma Physics

A new approach to the perturbative analysis of dynamical systems, which can be described approximately by soliton solutions of integrable nonlinear wave equations, is employed in the case of small-amplitude solutions of the ion acoustic wave equations of Plasma Physics. Instead of the traditional derivation of a perturbed KdV equation, the ion velocity is written as a sum of two components: elastic and inelastic. In the single-soliton case, the elastic component is the full solution. In the multiple-soliton case, it is complemented by the inelastic component. The original system is transformed into two evolution equations: An asymptotically integrable Normal Form for ordinary KdV solitons, and an equation for the inelastic component. The zero-order term of the elastic component is a single- or multiple-soliton-solution of the Normal Form. The inelastic component asymptotes into a linear combination of single-soliton solutions of the Normal Form, with amplitudes determined by soliton interactions, plus a second-order decaying dispersive wave. Satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and/or boundary conditions imposed on the inelastic component. The electrostatic potential is a first-order quantity. It is affected by the inelastic component only in second-order. The charge density displays a triple-layer structure. The analysis is carried out through third order.

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