Korteweg – deVries equation for longitudinal disturbances in coasting charged-particle beams

054402-1 This paper employs a one-dimensional kinetic model to investigate the nonlinear longitudinal dynamics of a long coasting beam propagating through a perfectly conducting circular pipe with radius rw. The average axial electric field is expressed as hEzi ebg0@ b=@z ebg2rw@ b=@z, where g0 and g2 are constant geometric factors, and b z; t R dpzFb z; pz; t is the line density. Assuming a waterbag distribution for the longitudinal distribution function Fb z; pz; t , it is shown that weakly nonlinear disturbances moving near the sound speed evolve according to the Korteweg–deVries equation.


I. INTRODUCTION
High energy accelerators and transport systems [1][2][3][4] have a wide variety of applications ranging from basic research in high energy and nuclear physics, to applications such as spallation neutron sources, heavy ion fusion, and medical physics, to mention a few examples. It is therefore important to develop an improved basic understanding of the nonlinear dynamics and collective processes in intense charged-particle beam systems. While there have been significant advances in three-dimensional numerical and analytical studies of the nonlinear Vlasov-Maxwell equations describing intense beam propagation, there is also considerable interest in the development and application of simplified one-dimensional kinetic models to describe the longitudinal dynamics of long coasting beams [5][6][7][8][9][10][11][12]. This paper employs a one-dimensional kinetic model recently developed by Davidson and Startsev [12] for a long coasting beam propagating through a perfectly conducting circular pipe with radius r w (Sec. II). The average axial electric field is expressed as hE s z i ÿe b g 0 @ b =@z ÿ e b g 2 r 2 w @ 3 b =@z 3 , where e b is the particle charge, the constants g 0 and g 2 are geometric factors that depend on the shape of the transverse density profile and location of the conducting wall, and b z; t R dp z F b z; p z ; t is the line density. Assuming a waterbag distribution [13][14][15][16] for the longitudinal distribution function F b z; p z ; t, it is shown that weakly nonlinear disturbances moving near the sound speed evolve according to the Korteweg-deVries (KdV) equation (Sec. III). The classical KdV equation [17][18][19][20][21] of course arises in several areas of nonlinear physics (e.g., hydrodynamics, plasma physics, etc.) in which there are cubic dispersive corrections to sound-wave-like signal propagation. The nonlinear KdV equation also has the appealing feature that it is exactly solvable [19,21] using inverse scattering techniques. Earlier treatments by Karpman et al. [22] and Bisognano [23] have also developed theoretical models leading to the Korteweg-deVries equation for weakly nonlinear disturbances in neutral plasma-loaded waveguides [22] and for intense beam propagation in a circular pipe [23]. An important difference between the present paper and these treatments [22,23] is that the present analysis provides a rigorous derivation of the KdV equation that makes use of the self-consistent kinetic model for longitudinal beam dynamics recently developed by Davidson and Startsev [12] that incorporates the important effects of transverse density profile shape, longitudinal beam thermal effects, etc. As such, the present analysis should have a wide range of applicability for weakly nonlinear disturbances moving slightly above the sound speed in intense charged-particle beams.
This paper considers longitudinal disturbances in a long coasting beam with characteristic radius r b . The beam is made up of particles with charge e b and rest mass m b propagating in linear geometry (the z direction) with directed axial kinetic energy b ÿ 1m b c 2 , where b 1 ÿ 2 b ÿ1=2 is the relativistic mass factor, V b b c is the average axial velocity of the beam particles, and c is the speed of light in vacuo. It is assumed that the beam propagates through a straight, perfectly conducting cylindrical pipe with wall radius r w , and the applied transverse focusing force F tr foc is modeled in the smoothfocusing approximation. Finally, the nonlinear dynamics of the beam particles is treated in the thin-beam (paraxial) approximation, and the particle motions in the beam frame are assumed to be nonrelativistic [1,12].

II. THEORETICAL MODEL AND ASSUMPTIONS
In the present analysis, we adopt a one-dimensional kinetic model recently developed by Davidson and Startsev [12] that describes the self-consistent nonlinear evolution of the longitudinal distribution function F b z; p z ; t, the average self-generated axial electric field hE z iz; t, and the line density b z; t R dp z F b z; p z ; t. For simplicity, the analysis is carried out in the beam frame (unprimed variables), and the beam intensity is assumed to be sufficiently low that the beam edge radius r b and rms radius R b hr 2 i 1=2 exhibit a negligibly small dependence on line density b . In addition, beam properties such as the number density n b r; z; t of beam particles are assumed to be azimuthally symmetric about the cylinder axis @=@ 0, where r; ; z are cylindrical polar coordinates with x r cos, y r sin, and r x 2 y 2 1=2 . Finally, the axial spatial variation in the number density n b r; z; t and line density b z; t 2 R r w 0 drrn b r; z; t is assumed to be sufficiently slow that k 2 z r 2 w 1, where @=@z k z L ÿ1 z is the inverse length scale of the z variation.
Within the context of these assumptions, the onedimensional kinetic equation describing the nonlinear evolution of the longitudinal distribution function F b z; p z ; t and the average axial electric field hE s z iz; t can be expressed correct to order k 2 z r 2 w in the beam frame as [12] @ @t where the geometric factors g 0 and g 2 are defined by Here, we have assumed a perfectly conducting cylindrical wall with E s z rr w 0 and consider the class of axisymmetric, bell-shaped density profiles n b r; z; t of the form In Eqs. (1) -(5), b R dp z F b 2 R r w 0 drrn b is the line density, r b is the edge radius of the beam (assumed independent of b ), and fr=r b is the profile shape function with normalization R 1 0 dXXfX 1=2. As a simple example, for fr=r b n 11 ÿ r 2 =r 2 b n , n 0; 1; 2; . . . , over the interval 0 r < r b , it can be shown that [12] In Eqs. (5) - (7), n 0 corresponds to a step-function density profile; n 1 corresponds to a parabolic density profile; and n 2 corresponds to an even more sharply peaked profile with n b rr b 0 @n b =@r rr b . Note from Eqs. (6) and (7) that the precise values of g 0 and g 2 exhibit a sensitive dependence on profile shape [12]. Moreover, the mean-square beam radius is R 2 b ÿ1 b 2 R r w 0 drrr 2 n b n 2 ÿ1 r 2 b for the choice of shape function fr=r b n 11 ÿ r 2 =r 2 b n , n 0; 1; 2; . . . . In any case, Eqs. (1) and (2) 1) and (2) of course depend on the form of the distribution function F b z; p z ; t. However, as a general remark, for small-amplitude perturbations (linearization approximation) Eqs. (1) and (2) support soundwave-like disturbances (with signal speed depending on g 0 and the momentum spread of F b ) with cubic dispersive modifications (depending on g 2 ) [12]. For present purposes, we specialize to the class of exact nonlinear solutions to Eq. (1) corresponding to the waterbag distribution [13][14][15][16] for ÿ1 < z < 1 (infinitely long coasting beam). Here, the distribution function F b A remains constant within the interval indicated in Eq. (8) and zero outside, whereas the boundary curves V ÿ b z; t and V b z; t, assumed single valued, distort nonlinearly as the system evolves according to Eqs. (1) and (2).
It is convenient to introduce the macroscopic fluid quantities corresponding to line density b R dp z F b , average axial velocity where v z p z =m b is the axial particle velocity. Some straightforward algebra that makes use of Eqs. (1), (2), and (8) gives the closed system of nonlinear fluid equations for b z; t, V b z; t, and P b z; t corresponding to Here, for the choice of waterbag distribution in Eq.
, and Q b 0 (exactly), which provides closure of the fluid equations. We therefore express where P b0 const and b0 const represent the unperturbed pressure and line density, respectively, and P b0 = 3 b0 1=12m b A 2 const. Here, A is the constant phase-space density in Eq. (8).

III. DERIVATION OF KORTEWEG -DEVRIES EQUATION
In the subsequent analysis we introduce the constant speeds U bT , U b0 , and U b2 defined by ; (13) and the normalized (dimensionless) fluid quantities z; t and Uz; t defined by b ÿ b0 b0 ; In Eqs. (13) and (14), U bT is the thermal speed, U b0 is the effective sound speed associated with the geometric factor g 0 , and U b2 is an effective speed that measures the strength of the cubic dispersive term in Eq. (10) associated with the geometric factor g 2 . Finally, it is convenient to introduce the scaled (dimensionless) time variables T and spatial variable Z defined by Making use of Eqs. (12) - (15), the nonlinear fluid description provided by Eqs. (9) -(11) reduces exactly to The fluid description provided by Eqs. (16) and (17) is exactly equivalent to the nonlinear kinetic description provided by Eqs. (1) and (2) for the choice of waterbag distribution in Eq. (8).
The fluid equations (16) and (17) in scaled variables are particularly amenable to direct analysis. For example, for traveling pulse (soliton) solutions we look for solutions to Eqs. (16) and (17) that depend on Z and T exclusively through the variable Z 0 Z ÿ MT, where M const is the normalized pulse speed measured in units of the sound speed U 2 b0 U 2 bT 1=2 . Making use of @=@T ÿM@=@Z 0 and @=@Z @=@Z 0 , Eq. (16) can be integrated once to give UZ 0 1 Z 0 MZ 0 , where use has been made of UZ 0 1 0 Z 0 1. Integrating Eq. (17) and substituting U M=1 then gives for Z 0 where 00 Z 0 1 0 and Z 0 1 0 are assumed. For present purposes, we solve Eq. (18) in the weakly nonlinear limit, treating jj 1 and retaining terms to order 2 . Introducing the scaled amplitudẽ Eq. (18) can be approximated by For M 2 > 1 the exact soliton solution to Eq. (20) is Consistent with the assumption of weak nonlinearity (small amplitude), in Eq. (21) it is assumed that M ÿ 1 , where 0 < 1, which corresponds to a compressional pulse > 0 moving slightly above the sound speed U 2 b0 U 2 bT 1=2 . In this case, the amplitude of the soliton in Eq. (21) is proportional to , whereas the soliton width is proportional to ÿ1=2 .