Soliton gas in space-charge-dominated beams

Based on the Vlasov–Maxwell equations describing self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. It has been shown that in the case of a beam with uniform phase-space density, the Vlasov–Maxwell equations can be replaced exactly by the hydrodynamic equations with a triple adiabatic pressure law coupled with the Maxwell equations. We further demonstrate that starting from the system of hydrodynamic and Maxwell equations, a set of coupled nonlinear Schrödinger equations for the slowly varying amplitudes of density waves can be derived. In the case where a parametric resonance between a certain mode of density waves and the external focusing occurs, the slow evolution of the resonant amplitudes in the cold-beam limit is shown to be governed by a system of coupled Gross–Pitaevskii equations. Properties of the nonlinear Schrödinger equation as well as properties of the Gross–Pitaevskii equation are discussed, together with soliton and condensate formation in intense particle beams.


Introduction
One of the main goals in the commissioning and operation of modern high-energy accelerators and storage rings is the achievement of higher and higher beam currents and charge densities. Hence, of particular importance are the effects of intense self-fields due to space charge and current, influencing the beam propagation, its stability and transport properties. In general, a complete description of collective processes in intense charged particle beams is provided by the Vlasov-Maxwell equations for the self-consistent evolution of the beam distribution function and the electromagnetic fields. Although the analytical basis (as mentioned above) for modelling the dynamics and behaviour of space-charge-dominated beams is well established, a thorough and satisfactory understanding of collective processes, detailed equilibrium and formation of patterns and coherent structures is far from being complete.
The present paper may be regarded as the first (to our knowledge) attempt to take a view of the description of the evolution and the collective behaviour of intense charged particle beams from an entirely different perspective, as compared with the ones available in the literature. We will be mainly interested in describing the slow evolution of some coarse-grained quantities that are easily measurable, such as the amplitudes of density waves. Owing to the nonlinear wave interaction contingent on the nonlinear coupling between the Vlasov and Maxwell equations, one can expect a formation of nontrivial coherent structure that might be fairly stable in space and time. Here, we show that solitary wave patterns in the beam density distribution are an irrevocable feature, characteristic of intense beams. Moreover, density condensates in the special case where a parametric resonance in terms of wave frequency between a particular mode of the fundamental density waves and the external focusing occurs, can be formed.
The paper is organized as follows. It was previously shown [1] that in the case of a sheet beam with constant phase-space density the Vlasov-Maxwell equations are fully equivalent to a hydrodynamic model with zero heat flow and triple-adiabatic equation-of-state. For consistency, in section 2, we repeat the derivation from a slightly different perspective and then use the hydrodynamic equations as a key for the subsequent analysis. In section 3, we consider the case of a smooth focusing where the time variation of the β-function (respectively, the time variation of the density envelope function) can be neglected. Furthermore, we demonstrate that, starting from the hydrodynamic equations, and using the renormalization group (RG) technique [2]- [5], a system of coupled nonlinear Schrödinger equations for the slowly varying amplitudes of density waves can be derived. The purpose of section 4 is twofold. First, we study the case where a parametric resonance between a particular mode of density waves and a resonant Fourier harmonic of the external focusing (a resonant harmonic from the √ β-function Fourier decomposition) occurs. It is shown that under certain conditions, the evolution of the resonant amplitudes of the forward and backward density waves is governed by a system of coupled Gross-Pitaevskii equations (see e.g. the review [6] and references therein). Secondly, it is demonstrated that in the non-resonant case the results obtained in the smooth focusing approximation coincide with the ones for a periodic focusing up to second order in the formal perturbation parameter. In section 5, the reduction of the infinite system of coupled nonlinear Schrödinger equations to a system of two coupled nonlinear Schrödinger equations is discussed. Such reduction is possible if one neglects the effect of all other modes and takes into account only the self-interaction of a single mode with a particular wave number. In the non-resonant case it is shown that the renormalized solution for the beam density describes the process of formation of holes (cavitons) in intense particle beams. The case where a parametric resonance is present is more interesting. In the cold-beam limit it is demonstrated that the evolution of the forward and the backward resonant wave amplitudes can be well described by a system of two coupled Gross-Pitaevskii equations. Finally, conclusions are drawn in section 6.

Derivation of the hydrodynamic model
We begin with the Hamiltonian describing the one-dimensional betatron motion in the presence of a space-charge field where e is the electron charge, R the mean radius of the accelerator, G(θ) the focusing strength of the linear machine lattice, V (x; θ) a contribution coming from nonlinear lattice elements (sextupoles, octupoles, etc) and E s and β s are the energy and the relative velocity of the synchronous particle, respectively. In addition, ϕ sc (x; θ) is the self-field potential due to spacecharge, which satisfies the one-dimensional Poisson equation Here n = N p /V t is the density of beam particles (N p is the number of particles in the beam, and V t the volume of the area occupied by the beam in the transverse direction), 0 the dielectric susceptibility and f(x,p; θ) the distribution function in phase space. Equations (2.1) and (2.2) can be written in an alternative form as is the beam perveance and ϕ sc = eN pφ / 0 . Note that the beam perveance λ is a quantity with dimension m −1 , whereas the dimension ofφ is m 2 . In addition, the one-particle distribution function f(x,p; θ) entering the integral on the right-hand side of equation ( For simplicity, we will omit the tilde sign in what follows. Next, we perform a canonical transformation defined by the generating function where α and β (and γ) are the well-known Twiss parameters satisfying the equations As a result, we obtain the new Hamiltonian where the self-field potential U(x; θ) (φ = √ βU ) satisfies the equation is the derivative of the phase advance with respect to θ.
We are now ready to write the Vlasov equation for the one-particle distribution function f (x, p; θ) in the two-dimensional phase space (x, p). It reads as and f(x, p; θ) = 0 otherwise. The proof that such a solution to the Vlasov equation (2.13) exists can be found in the appendix. The equations for the boundary curves p (±) (x; θ) can be derived as follows. We substitute the explicit form (2.15) of the distribution function into the Vlasov equation (2.13). Here H(z) is the well-known Heaviside function. Using the fact that the derivative of the Heaviside function with respect to its argument equals the Dirac δ-function, we obtain Multiplying equation (2.16) first by 1 and then by p, and integrating the result over p, we readily obtain where ρ(x; θ) and v(x; θ) are the density and the current velocity, respectively. From the two definitions above, it follows that v = 1 2 (p (+) + p (−) ).

(2.22)
Defining further the pressure P(x; θ) and the heat flow Q(x; θ) and using equation (2.22), we have From equation (2.20), it follows that the pressure can be expressed as (2.25) In addition, it is straightforward to verify that Using the continuity equation (2.27), we cast the system of equations (2.27)-(2.29) in its final form which will be the starting point for the subsequent analysis. Here is the normalized thermal speed-squared, and τ = χ(θ) + τ 0 is the new independent 'time' variable.

Renormalization group reduction of the hydrodynamic equations
Before proceeding further, we make an important remark. Let F(θ) be a periodic function of θ with period 2π. Noting that the phase advance χ(θ) can be represented in the form where ν is the betatron tune and χ p (θ + 2π) = χ p (θ), we can expand F(θ), regarded as a function of χ (resp. τ) in a Fourier series in the new variable χ (resp. τ), as follows: The expansion coefficients A n are given by the expressions Using the definition of the phase advance (3.1), and choosing τ 0 = χ p (π), we can rewrite the Fourier expansion (3.2) as where the expansion coefficients B n are given by the expressions In what follows, we consider the case where the external potential V is zero, so that the set of equations to be analysed acquires the form Furthermore, we assume that there exists a nontrivial solution to equations (3.6)-(3.8) in the interval x ∈ (−x (−) , x (+) ), and that the sheet beam density is zero ( = 0) outside of the interval. We introduce also the ansatz where the envelope function E(τ) is a solution to the equation and therefore can be represented in explicit form as b n e inτ/ν , b n = a n 1 − n 2 /ν 2 .
(3.13) It enables us to rewrite equations (3.6)-(3.8) as follows: Next, we differentiate the first equation (3.14) of the above system with respect to τ and the second equation (3.15) with respect to x. After summing these up, we obtain Equation (3.16) supplemented with equation (3.14) rewritten as comprises the basic system of equations for the analysis in the subsequent exposition.
In what follows, we consider the case of smooth focusing, where the time variation of g(τ) [E(τ), resp.] can be neglected. In the next section, we will show that when the frequencies of all fundamental modes are sufficiently far from a parametric resonance, this assumption holds true to second order in the perturbation parameter even in the case when such time variation is present. Thus, the basic equations (3.16) and (3.17) acquire the form Let us further assume that the actual dependence of R and u on the independent variables is given by is a slow spatial variable. Therefore, the basic equations (3.18) and (3.19) can be rewritten as Following the basic idea of the RG method, we represent the solution to equations (3.23) and (3.24) in the form of a standard perturbation expansion [7] in the formal small parameter as The zeroth-order equations (3.23) and (3.24) read as Their solutions can be found in a straightforward manner to be Here A m and B m are constant complex amplitudes (to this end, dependent on the slow variable ξ only) of the backward and the forward wave solutions to equation (3.26), respectively. These will be the subject of renormalization at the final step of the renormalization group procedure resulting in RG equations governing their slow evolution. Furthermore, The discrete mode frequencies ω m are determined from the dispersion relation It can be easily verified that the above choice of parameters leads to which means that linear perturbation to the uniform density ρ 0 = E −1 0 average to zero and do not affect the normalization properties on the interval (−x (−) , x (+) ). In addition, the following conventions and notations have been introduced. The first-order perturbation equation for R 1 reads as where The solution for R 1 is readily obtained to be where Note that the (infinite-dimensional) matricesα (±) are symmetric, i.e.
Having determined R 1 (x; τ), the first-order current velocity u 1 (x; τ) can be found in a straightforward manner. The result is where Note that the (infinite-dimensional) matrixβ (+) possesses the same symmetry properties as those displayed by equation (3.39) possessed by the matrixα (+) , whereasβ (−) is antisymmetric (and evidently β (−) mm = 0). In second order, the equation for R 2 (x, ξ; τ) acquires a form similar to that of equation (3.34). It is important to emphasize that this assertion holds true in every subsequent order. Each entry on the right-hand sides of the corresponding equations can be calculated explicitly utilizing the already determined quantities from the previous orders. The righthand side of the equation for R 2 (x, ξ; τ) contains terms which yield oscillating contributions with constant amplitudes to the solution for R 2 (x, ξ; τ). Apart from these, there are resonant terms (proportional to e iz (±) m (x;τ) ) leading to a secular contribution. To complete the renormalization group reduction of the hydrodynamic equations, we select these particular resonant secondorder terms on the right-hand side of the equation determining R 2 (x, ξ; τ). The latter can be written as

The parametric resonance
Let us now address the system of equations (3.16) and (3.17). Without loss of generality, we assume that the time variation of g(τ) can be treated as a second-order perturbation (which is usually the case), i.e. g(τ) = a 0 + 2 n =0 a n e inτ/ν . (4.1) The same holds true for the envelope function E(τ) b n e inτ/ν , b n = a n 1 − n 2 /ν 2 . (4.2) Thus, the basic equations (3.16) and (3.17) can be rewritten in the form Let us reiterate that the assumption concerning the second order of magnitude in of the time variation of g(τ) and E(τ) does not restrict the generality of the subsequent analysis. If this variation were of first order in , the proper perturbation parameter to use would be √ instead of . In addition, the variables R and u have to be rescaled accordingly, Although the general case can, in principle, be treated through more labour-intensive manipulations, for the sake of simplicity in what follows, we select a particular mode with mode number m. The zeroth-order solution can be written in the form x,τ) ] + c.c., (4.9) where again The particular mode is chosen such that an exact parametric resonance (if possible) 2ω = n 0 /ν (4.10) occurs for some integer n 0 . The first-order perturbation equation for R 1 can be written in the form Here implies exclusion of the harmonic with n = n 0 from the sum. It is important to note that terms giving rise to the secular contribution due to the parametric resonance vanish identically in the first order. The solution for R 1 can be written explicitly as AB * e i(z (+) −z (−) ) + c.c. (4.12) Straightforward calculations yield the solution for u 1 In second order, we take into account terms providing the secular contribution to the solution only. Thus, we write the equation for R 2 as (4.14) Similar to equation (3.45), we obtain in a straightforward manner Collecting once again the zeroth-order term together with all secular terms in higher orders and renormalizing the amplitudes A and B, we obtain the RG equations the index m and the tilde sign over the new amplitudeB has been omitted. Clearly enough, equations (5.1) and (5.2) follow directly from the system (4.16)-(4.17) if b n 0 is set to zero. This implies that in the case where a parametric resonance does not occur in the original system, the smooth approximation is valid to second order in the formal perturbation parameter. Moreover, it is straightforward to verify that G is always positive. The system of two coupled nonlinear Schrödinger equations (5.1) and (5.2) is in general non-integrable. Its integrability is proven by Manakov [8] only in the simplest case of G = −1.
If one of the amplitudes (B or A) in its capacity of being a particular solution to the corresponding nonlinear Schrödinger equation is identically zero, the equation for the other amplitude (say A) becomes In nonlinear optics, equation (5.4) is known to describe the formation and evolution of the so-called dark solitons [9]. In the case of charged particle beams these correspond to the formation of holes or cavitons in the beam. The solution to equation (5.4) can be written as for all c and provided n > c 2 /2.
Recently, Koscielniak et al [10] reported formation of holes in the longitudinal density profile which has been predicted by computer simulations and observed at the CERN PS Booster and at Fermilab. Although it is not straightforward to compare our results with the findings of Koscielniak et al, it is worth noting that regardless of the case (longitudinal or transverse phase space) the nonlinear interaction between density waves in space-chargedominated beams as a rule leads to the formation of holes in the density profile. This is apparently due to the repulsive character of the electromagnetic interaction between individual particles within the beam.
Let us now examine equations (4.16) and (4.17). In the cold-beam limit v T → 0 (or equivalently ω → 1) the second terms on their right-hand sides can be neglected as compared with the other terms. Therefore, we can write where the local gauge transformatioñ has been performed on the amplitudes A and B. Equations (5.9) and (5.10) represent a system of two coupled Gross-Pitaevskii equations, known to govern the formation of condensate structures in atomic gases confined in magnetic traps. In addition, superfluidity in helium as well as patterns in the gas of paraexcitons in semiconductors are considered as a possible manifestation of Bose-Einstein condensation. It is remarkable that under certain conditions similar phenomenon can be observed in space-charge-dominated beams.
To conclude the present section, we would like to point out that the nonlinear Schrödinger equation provides a canonical description for the coarse-grained envelope dynamics of a quasi-monochromatic plane wave propagating in a weakly nonlinear dispersive medium when dissipative processes are absent or negligibly small. It arises in various physical contexts and governs the evolution of nonlinear waves such as propagation of a laser beam in a medium whose index of reflection is sensitive to the wave amplitude, water waves at the free surface of an ideal and plasma waves. The nonlinear Schrödinger equation also appears in the description of the Bose-Einstein condensate, a context where it is often called the Gross-Pitaevskii equation. It admits solutions in the form of coherent structures like vortices which define states that can be excited in super helium. The last few years have witnessed a rapid development in the research of nonlinear-Schrödinger-equation-related applications such as optical communications, laser surgeries, remote sensing and the Bose-Einstein condensation. It is also remarkable that such applications can be found in the physics of beams.

Concluding remarks
The analysis performed in the present paper is based on the Vlasov-Maxwell equations for the self-consistent evolution of the beam distribution function and the electromagnetic fields. We considered the propagation of an intense beam through a periodic focusing lattice. It has been proven that a special class of solutions to the Vlasov-Maxwell equations exists, which defines a uniform phase-space density inside of simply connected boundary curves. These solutions provide an exact closure to the hierarchy of moments resulting in a set of hydrodynamic equations for the beam density (continuity equation) and for the current velocity. The latter is characterized by a triple adiabatic pressure law. The major drawback of the model discussed here is that due to the constancy of the distribution function inside of the evolving region of phase space, it fails to take into account an important feature such as the well-known Landau damping. Inclusion of the Landau damping in the description can be achieved correctly only by direct analysis of the Vlasov-Maxwell system [11].
Further, we studied first the case when the smooth focusing approximation applies. Based on the RG method, a system of coupled nonlinear Schrödinger equations has been derived for the slowly varying amplitudes of interacting beam-density waves. Under the approximation of an isolated mode neglecting the effect of the rest of the modes, this system reduces to two coupled nonlinear Schrödinger equations for the amplitudes of the forward and of the backward wave. The particular solution to the latter system asserting that the amplitude of either of the waves (the forward or the backward) can vanish identically, leads to a single nonlinear Schrödinger equation for the other amplitude. The latter is characterized by a repulsive nonlinearity and, therefore, it describes the evolution of holes in intense charged particle beams.
We would like to reiterate that regardless of the case (longitudinal or transverse phase space) the nonlinear interaction between density waves in space-charge-dominated beams as a rule leads to the formation of holes in the density profile. This is apparently due to the repulsive character of the electromagnetic interaction between individual particles within the beam. However, one can expect this situation to change drastically in the cases of partially neutralized beams such as counter-rotating electron beam cooling the main beam of positively charged particles, electron cloud around the accelerated beam of positively charged particles, etc. It remains to study these cases using the methods developed here, which will be the subject of a future work.
The analysis of periodic focusing clearly showed that the results obtained in the case of smooth focusing remain unchanged up to second order in the perturbation parameter if a parametric resonance between a particular mode and an appropriate Fourier harmonic of the external focusing does not occur. If however, an exact parametric resonance takes place, it was shown that the evolution of the wave amplitudes of the resonant mode in the cold-beam limit is described by a system of coupled Gross-Pitaevskii equations. Quite remarkably, it was found that there exists a possibility of formation of density condensates in space-chargedominated beams.
As a matter of fact, equation (A.5) is equivalent to the two equations ∂p (±) ∂θ with the initial conditions (A.4), provided U is defined as a solution to equation (2.19).