PDF Evolution of two-dimensional lump nanosolitons for the Zakharov ... - UNAM

Evolution of two-dimensional lump nanosolitons

for the Zakharov-Kuznetsov and electromigration equations

M. C. Jorge^a兲 and Gustavo Cruz-Pacheco^b兲

FENOMEC, Department of Mathematics and Mechanics, I.I.M.A.S.,

Universidad Nacional Autónoma de Mexico, Apdo. 20-726, 01000 México, D.F.

Luis Mier-y-Teran-Romero^c兲

Department of Physics, Northwestern University, Evanston, Illinois, 60208 Noel F. Smyth^d兲

School of Mathematics, The King’s Buildings, University of Edinburgh, Edinburgh, EH9 3JZ Scotland, United Kingdom

共Received 14 October 2004; accepted 2 February 2005; published online 21 October 2005兲 The evolution of lump solutions for the Zakharov-Kuznetsov equation and the surface electromi- gration equation, which describes mass transport along the surface of nanoconductors, is studied.

Approximate equations are developed for these equations, these approximate equations including the important effect of the dispersive radiation shed by the lumps as they evolve. The approximate equations show that lumplike initial conditions evolve into lump soliton solutions for both the Zakharov-Kuznetsov equation and the surface electromigration equations. Solutions of the approxi- mate equations, within their range of applicability, are found to be in good agreement with full numerical solutions of the governing equations. The asymptotic and numerical results predict that localized disturbances will always evolve into nanosolitons. Finally, it is found that dispersive radiation plays a more dominant role in the evolution of lumps for the electromigration equations than for the Zakharov-Kuznetsov equation. © 2005 American Institute of Physics.

关DOI:10.1063/1.1877892兴

The phenomenon of electromigration plays a major role in nanocircuits in electronics as it leads to undesirable surface instabilities. In order to suppress or delay the onset of these instabilities it is necessary to have a quan- titative understanding of them. A step in this direction is to model the surface of a conductor as a free surface with appropriate boundary conditions. Under the assumption of unidirectional waves which have small lateral disper- sion, it is shown that the instabilities are governed by a Zakharov-Kuznetsov (ZK) type equation for the surface waves, coupled to an elliptic equation for the electrostatic potential. This coupled system of equations is noninte- grable and numerical solutions show that disturbances propagate as lump solitons in two spatial dimensions.

Moreover, as a lump propagates, it sheds radiation and it is found that this radiation plays a dominant role in the evolution of the lump. In the present work the evolution of lump disturbances is studied asymptotically and nu- merically. It is shown by using a combination of an ap- proximate theory (collective coordinates) for the lump evolution, coupled to a geometric optics solution for the shed radiation, how an initial lump evolves by shedding radiation into a traveling, stable, coherent lump struc- ture. It is shown that all initial conditions ultimately

evolve into such lumps. This evolution is also studied for higher order nonlinearity and it is shown that, despite the shed radiation, an initial lump collapses in finite time if it has a large enough initial amplitude. Finally, due to the simplicity of the equations and the shed radiation pat- tern, we are able to extend the theory to the full three space dimensional case. In this full three-dimensional case, similar behavior as in two spatial dimensions is found. The present work therefore suggests that instabili- ties always form and keep their coherence over long dis- tances. Thus in order to eliminate instabilities in nanocir- cuits, it is necessary that the surface of the conductor is coupled to a mechanism capable of producing a threshold for the instability, such as a surface coating.

I. INTRODUCTION

In the present work the evolution of coherent structures for the Zakharov-Kuznetsov 共ZK兲 equation will be consid- ered. The ZK equation describes small amplitude, nonlinear waves propagating in two spatial dimensions.¹ The deriva- tion of this equation assumes that the propagation of the waves is fundamentally unidirectional, as for the one space dimensional Korteweg–de Vries共KdV兲equation, and that the waves have a relatively weak lateral dispersion. The ZK equation, like the KP equation, is a generalization of the KdV equation to two space dimensions which takes account of weak lateral dispersion.

In previous work the linear stability of the planar KdV soliton, which is a solution of the ZK equation, with respect

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d兲Author to whom correspondence should be addressed. Electronic mail:

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to long wave transverse perturbations was studied. It was shown in Refs. 2–4 that for long transverse wave perturba- tions the KdV soliton is unstable. This instability was fol- lowed numerically in Ref. 5 and it was observed that an initially transversely uniform soliton breaks up into several lumplike structures. These structures were identified as cy- lindrical solitarylike waves.

More recently the ZK equation coupled to a nonlinear Laplace-type equation has been obtained in the study of sur- face electromigration in nanoconductors subject to strong, unidirectional electric fields.^6,7 In this situation the surface electromigration of electrons deforms the conductor. In the early stages, this deformation can be identified with a ZK lump solitary wave. In many other applications, a ZK-type equation arises, but with a different order nonlinearity,⁸ so that the resulting equation is a possible two-dimensional counterpart of the mKdV equation.⁹

In the present work the evolution of an initial condition close to a lump soliton for the ZK equation will be studied.

As in previous work for the KP equation,¹⁰ this evolution is studied using an approximate theory, which couples the lump soliton to the dispersive radiation shed as it evolves. The present work considers the general cases of a ZK equation with an arbitrary order nonlinearity and the ZK equation coupled to a potential equation, as arising in electromigration in nanoconductors. It is found that lumplike initial conditions evolve into steadily propagating Gaussian soliton solutions by shedding mass in the form of dispersive radiation. It is further found that, as expected, this shed dispersive radiation plays a critical role in the stabilization of the soliton. A major difference between the ZK equation and the electromigration equations is that the shed radiation has a greater effect on the evolution of the pulse in the case of the electromigration equations, with the radiation carrying a significant amount of momentum. When the nonlinearity in the ZK equation is higher than a quadratic, the approximate equations show that initial conditions with mass below a certain threshold decay, while those with mass above the threshold blow up in finite time. Finally, it is found that there is very good agreement between full numerical solutions of the ZK equation and solutions of the approximate equations, which include the effects of shed radiation.

The paper is organized as follows. In the second section the basic ZK equations in two and three space dimensions are formulated. In the third section, approximate equations for a lump solution of the single ZK equation are derived, these equations also taking account of the dispersive radia- tion shed as the lump evolves. The fourth section extends the approximate equations for the ZK equation to three space dimensions. In the fifth section, corresponding approximate equations for the ZK system governing electromigration are derived with the sixth section presenting comparisons be- tween full numerical solutions of the governing equations and solutions of the approximate equations. Finally, the last section presents the conclusions.

II. FORMULATION

The ZK equation¹in nondimensional variables is

⳵^u

⳵^{t + 6u}

⳵^u

⳵^x+

⳵^3u

⳵^x3+

⳵^3u

⳵^x⳵^{y2= 0. 共1兲}

This equation is subject to a general initial condition u共x,y, 0兲=f共x,y兲. Herexis the coordinate in the direction of propagation of the soliton andyis the direction transverse to this. The amplitude of the disturbance isuandt is the time variable. The last term in the ZK equation共1兲accounts for weak lateral dispersion. The modified ZK equation with a general nonlinearity is obtained from the ZK equation by replacingu in the convective term byu^p.

The coupled system of the ZK equation and a potential equation governing electromigration in nanoconductors⁷is

⳵^u

⳵^{t + 6u}

⳵^u

⳵^x+

⳵^3u

⳵^x3+

⳵^3u

⳵^x⳵^y2+ 1 2

冉

^⳵⳵^ux

⳵⌿

⳵^{x +}

⳵^u

⳵^y

⳵⌿

⳵^y

冊

^{= 0,}

共2兲

⳵²⌿

⳵^{x2 +}

⳵²⌿

⳵^{y2 =}

⳵^u

⳵^{x, 共3兲}

together with the initial conditions u共x,y, 0兲=f共x,y兲 and

⌿共x,y, 0兲=共ⵜ²兲⁻¹fx共x,y兲. In this application the variableuis the surface displacement and⌿ is the electrostatic potential on the surface of the conductor. The ZK equation is then modified by the convective termⵜu·ⵜ⌿. The electrostatic potential ⌿ is in turn determined by the motion of the free surface.^6,7

The ZK equation共1兲with a general nonlinearity⁸ has a straightforward counterpart in three spatial dimensions

⳵^u

⳵t + 6u^p⳵^u

⳵^x+

⳵

⳵x

冉

^⳵⳵²x^u2+⳵^2u

⳵y²+⳵^2u

⳵z²

冊

^{= 0, 共4兲}

together with the initial conditionu共x,y,z, 0兲=f共x,y,z兲. The nonlinear powerpmust be a positive integer sinceu can be negative.

In the present work the evolution of ufor the ZK equa- tions 共1兲 and 共4兲 and the ZK system 共2兲 and 共3兲 will be analyzed for the specific initial conditions

f共x,y兲=Ae^{−␬共x2+y2兲} and f共x,y,z兲=Ae^{−␬共x2+y2+z2兲}. 共5兲 Finally, it should be noted that the ZK equations共1兲,共2兲, and共4兲satisfy the mass conservation equation

d

dt

冕

_−⬁^{⬁ u dV= 0 共6兲}

and the momentum conservation equation d

dt

冕

−⬁

⬁

u²dV= 0, 共7兲

wheredVis appropriately interpreted asdxdyin two dimen- sions and dxdydz in three dimensions. These conservation equations follow from the ZK equations foruon integrating by parts.

Numerical solutions of the ZK equation 共1兲 for the pa- rameter values A= 0.4 and ␬= 0.05 for the initial condition 共5兲 and the electromigration equations 共2兲 and 共3兲 for the parameter valuesA= 0.3 and␬= 0.05 for the same initial con-

dition are displayed in Figs. 1 and 2. In Fig. 1共a兲the numeri- cal solution of the ZK equation共1兲att= 10 is shown, with a cross section through y= 0 at this same time shown in Fig.

1共b兲. A shelf of radiation behind the evolving lump can be clearly seen, although the shelf has a variation in they di- rection which will not be accounted for in the present work.

Figure 1共c兲shows the numerical solution of the ZK equation

at the later timet= 25. It is clear that the lump initial condi- tion has evolved into a lump soliton followed by a tail of radiation within a caustic. This radiation propagates to the left, as expected since the group velocity for waves for the linearized ZK equation is negative. The lump eventually settles to a steady traveling wave after radiating away its excess mass and momentum. In addition it can be seen that at least one more soliton is developing from the radiation behind the initial lump. Very similar evolution for the coupled system of the ZK equation 共2兲 and the potential equation共3兲is observed in Fig. 2. The conclusion is that the surface displacementu drives the potential⌿, which in turn modifiesu. However, this modification is just in the velocity ofu, due to the convective term. In the next section we shall turn to the derivation of the approximate equations for the ZK equation for the evolution of the initial condition共5兲.

III. APPROXIMATE EVOLUTION EQUATIONS Let us first consider the modified ZK equation

⳵^u

⳵^{t + 6u}

p⳵^u

⳵^x+

⳵^3u

⳵^x3+

⳵^3u

⳵^x⳵^{y2= 0, 共8兲}

where againp must be a positive integer, together with the initial condition

u共x,y,0兲=Ae^{−␬共x2+y2兲}. 共9兲 In order to derive the approximate evolution equations for this initial condition, we proceed as in Refs. 10 and 11 by setting

u共x,y,t兲=u₀+u₁=a共t兲e^{−␬共t兲关共x−␨共t兲兲2+y2兴}+u₁. 共10兲 The position of the soliton is then共␨共t兲, 0兲, its amplitude is a共t兲, and its width is␬⁻¹共t兲. The function u₁ represents the radiation shed by the pulse as it evolves. The approximate equations governing the evolution of the pulse are now ob- tained by an appropriate use of the conservation equations 共6兲and共7兲, resulting in equations of motion fora,␬^{, and}␨^. It is remarked that the procedure of Refs. 10 and 11 for the

FIG. 1. Numerical solution of the ZK equation共1兲for the initial condition 共5兲withA= 0.4 and␬^{= 0.05.}共a兲Solution att= 10.共b兲Section of solution alongy= 0 att= 10.共c兲Solution att= 25.

FIG. 2. Numerical solution of the electromigration equations共2兲and共3兲at t= 30 for the initial condition共5兲withA= 0.3 and␬= 0.05.

KP and KdV equations, respectively, is a version of the modulation theory of Whitham¹² whereby the spatial modu- lations of the trial functions, which are not exact solutions, are neglected, but the relevant degrees of freedom of the shed radiation are included. Equations for the collective co- ordinates for both the lump and the shed radiation are then obtained.

As can be seen from Figs. 1共a兲and 1共c兲the lump partu₀ of u is separated from the radiationu₁ and the radiation is confined to a region bound by caustics. First the region of the radiation between the caustics then needs to be determined, as in Ref. 10. In order to determine this region, we again observe from Fig. 1 that兩u1兩Ⰶ兩u0兩. The equation governing the radiation is therefore the linearized ZK equation

⳵^u1

⳵^{t +}

⳵^3u1

⳵^{x3 +}

⳵^3u1

⳵^x⳵^y2=g共x−␨共t兲,y兲, 共11兲

together with the initial conditionu₁共x,y, 0兲= 0. The termg on the right-hand side represents the source of radiation due to the pulseu₀. This source term is assumed to have small extent relative to the extent of the linear radiationu₁.

Assuming that ␨ is slowly varying, the solution of the radiation equation共11兲is

u₁共x,y,t兲= 1 共2␲兲²

冕

−⬁

⬁

冕

−⬁

⬁

U共k,l兲e^{i␸共k,l,x,y,t兲}dkdl, 共12兲 where the phase␸ ^is ␸⁼k共x−␨兲+ly−␻共k,l兲t and the linear dispersion relation is␻^{= −k3−kl2}. The kernel U共k,l兲 is re- lated to the source of radiationg. The actual form ofUplays no role in the determination of the region of confinement of the radiation.

As for general linear dispersive wave theory,¹²the radia- tion is confined inside the caustics of共12兲. These caustics are determined by finding the evolution ofu₁for large time us- ing the method of stationary phase. The points of stationary phase for the integral共12兲are given by the solutions of

0 =⳵␸

⳵^{k =x−}␨⁺共3k²+l²兲t,

共13兲 0 =⳵␸

⳵^{l =y+ 2klt.}

Solving forlusing the second of these equations and substi- tuting into the first gives the curves of constant wave number kas the family of parabolas

x−␨⁺

冉

^3k2+4ky22t2

冊

^{t= 0. 共14兲}

The envelope of this family of parabolas is the curve of coalescence of stationary points, which is the caustic. To calculate the envelope, Eq. 共14兲 and its derivative with re- spect to the parameterkare solved simultaneously fork. This gives the equation for the caustic as

x−␨共t兲±

冑

3y= 0. 共15兲

The caustic is therefore straight lines forming a wedge with apex at the lump located at共␨共t兲, 0兲and which trails behind it.

It should be noted that, unlike the situation for the KP equation,¹⁰ the caustics do not change with time. The shed radiation is confined to the region between the caustics. This geometric argument then explains the radiation patterns in Fig. 1共c兲. Exactly the same calculation of the caustic is pos- sible for the three space dimensional ZK equation. In this three-dimensional case, the caustic is a conical surface. The three-dimensional case will be taken up further in Sec. V.

With this information about the location of the radiation, we can now determine the approximate equations for the param- eters of the approximate lump solution共10兲.

Let us begin by determining an approximate lump solu- tion of permanent form for the ZK equation. To this end we seek solutions of the form

u=v共x−ct,y兲. 共16兲

Substituting this form into the modified ZK equation共8兲, it can be found, after an integration in x, that v satisfies the elliptic partial differential equation

−cv+ 6

p+ 1v^p+1+ ⳵^2v

⳵␩²⁺

⳵^2v

⳵^{y2= 0, 共17兲} where ␩^=x−ct. Now this elliptic equation is the Euler- Lagrange equation for the functional

F共v兲=

冕

−⬁

⬁

冕

_−⬁^⬁

冋

^{12共v␩2+vy2兲−共p+ 1兲共6p+ 2兲vp+2}

+c

2v²

册

^{d␩dy. 共18兲}

At this stage the stability of the two-dimensional solitons could be analyzed following Refs. 13–15. To this end, one could use a trial function in the functional共18兲which is the exact, but analytically unknown, soliton solution scaled to conserve momentum. The variation of the functional 共18兲 evaluated along this scaled soliton will then give the stability region in parameter space, as in Refs. 13–15. However, in the present work, we are interested in not just the stability, but the dynamics of the approach to equilibrium, or depar- ture from it. Hence we shall not proceed as in Refs. 13–15, but assume a trial function as in Ref. 11. However, a brief outline of the main ideas will now be given.

The functional in 共18兲 is just the Hamiltonian with the momentum constrained with the velocityc as the Lagrange multiplier.^14,15Therefore, to obtain a rigorous assessment of the stability of the two-dimensional soliton, we proceed as in Refs. 13–15. Letus共x,y兲be the exact extremum of the func- tional 共18兲 constrained to the value P₀ of the momentum.

Then consider the curve, in function space, fa共x,y兲=1

aus共x/a,y/a兲. 共19兲

This curve is the soliton solution whena= 1 and satisfies the momentum constraint since

冕

−⬁

⬁

冕

−⬁

⬁

f_a²dxdy=

冕

−⬁

⬁

冕

−⬁

⬁

u_s²dxdy=P₀. 共20兲 Evaluation of the functional共18兲along the curvef_agives

F共fa兲= T a²− U

a^p+cP₀=H共a兲+cP₀, 共21兲 whereH共a兲is the Hamiltonian evaluated along the curve and

T=1 2

冕

−⬁

⬁

冕

−⬁

⬁

兩ⵜus兩²dxdy and

共22兲

U= 6

共p+ 1兲共p+ 2兲

冕

_−⬁^⬁

冕

_−⬁^{⬁ usp+2dxdy.}

Clearly, by construction, H共1兲 is an extremum of the Hamiltonian. The study of the nature of the Hamiltonian along the curvefaamounts to a study of the stability of the critical point ata= 1. To this end

H⬘共a兲= −2T a³ + pU

a^p+1, 共23兲

which gives H⬘^{共1兲= 0 as 2T=pU at a}= 1. This also shows that the potential energy U is positive. The nature of the critical point is then determined by

H⬙共1兲=T关6 − 2共p+ 1兲兴. 共24兲

From the second derivative 共24兲 it is clear that along the curve fa the soliton solution is a minimum for p⬍2 and a maximum forp⬎2. The behavior at the critical valuep= 2 is not determined by this simple analysis and appears as a threshold in a similar way to other soliton equations treated in Refs. 14 and 15.

This analysis following Refs. 14 and 15 provides a rig- orous result for the instability of the two-dimensional soliton.

In the stable case it is not conclusive as all directions must be included to determine stability. However, the analysis pro- vides a rigorous result in a restricted class of deformations in the stable case. Since in the present work we are interested in the study of the dynamics of the evolution of lump-type ini- tial conditions, we shall now focus on determining approxi- mate equations for the collective coordinates for both an ap- proximate lump solution and the shed radiation which plays the essential role in stabilizing the lump solution of this in- finite dimensional Hamiltonian system.

Let us now assume that the trial functionvfor the trav- eling wave solution共16兲is the Gaussian

v=ae^{−␬共␩2+y2兲}. 共25兲 Substituting this trial function into the Lagrangian共18兲, the average Lagrangian can be found to be

F共a,␬^,c兲=␲

2a²− 6␲ 共p+ 1兲共p+ 2兲²

a^p+2

␬ ⁺

␲^c 4

a²

␬^{. 共26兲} The variational equations Fa and F_␬ for this averaged La- grangian then have a stationary point given by

c= 12

共p+ 1兲共p+ 2兲a^p− 2␬ and a^p=共p+ 1兲共p+ 2兲²

6p ␬.

共27兲 These equations give the velocity and amplitude of the fixed point, which is an approximate steady solitary wave solution of the modified ZK equation. It can be found that the velocity-amplitude relation for the solitary wave is

c= 24

共p+ 1兲共p+ 2兲²a^p. 共28兲

This relation shows that there is a one-parameter family of approximate steady solitary wave solutions.

Using these results, the approximate equations for the solitary wave can now be determined. The first approximate equation is just the solitary wave velocity

␨^˙=V=c共a兲, 共29兲

wherecis given by 共28兲. The remaining approximate equa- tions fora and␬are found by using the conservation equa- tions共6兲and共7兲. The momentum conservation equation共7兲 gives

0 = d dt

冕

−⬁

⬁

冕

−⬁

⬁

u²dxdy= d dt

冕

−⬁

⬁

冕

−⬁

⬁

共u₀²+ 2u₀u₁

+u₁²兲dxdy. 共30兲

Now, as can be seen from the numerical solutions of Fig. 1 and has been determined from the caustic derivation above, the radiationu₁does not overlap the solitonu₀. Furthermore, as the radiation has small amplitude relative to the pulse,u₁² is negligible. Then, using the trial function共10兲 and the ve- locity expression共29兲, we have

d

dt

冕

_−⬁^⬁

冕

_−⬁^{⬁ u02dxdy=dtd}

冕

_−⬁^⬁

冕

_−⬁^{⬁ a2e−2␬关共x−␨兲2+y2兴dxdy}

= d dt

␲^a2

2␬ ^{= 0. 共31兲}

The equation for conservation of mass will then close the system of approximate equations共29兲and共31兲.

Let us consider mass balance in the shaded regionR共t兲 shown in Fig. 3, which is the region outside the caustic. That is we calculate

d dt

冕

R共t兲

u dxdy. 共32兲

Differentiating we obtain d

dt

冕

R共t兲

u dxdy=

冏

^{− 2V}

^冕

^0⬁u

冏

x=␩−冑3y

dy

−

冕

R共t兲

⳵

⳵^x

冋

^{p6+ 1up+1+uxx+uyy}

册

^dxdy.

共33兲 Since the regionR共t兲 does not include, by construction, the radiation,ucan be replaced in this equation by the Gaussian

lump in 共10兲. The following expressions are then obtained for the various integrals in共33兲:

冕

R共t兲

u dxdy=5␲ 6

a

␬^,

冏 ^冕

^0⬁u

冏

x=␩⁻冑3y

dy=

冑

_␲

4

冑

a_␬^{, 共34兲}

冕

R共t兲

⳵

⳵^x

冋

^{p+ 16 up+1+uxx+uyy}

册

^dxdy

=

冏

^{− 2}

^冕

^0⬁

^冋

^{p6+ 1up+1+uxx+uyy}

^册 冏

x=␩⁻冑3y

dy

= − 2

冑

_␲

冉

4共p+ 1兲^6ap+13/2␬^1/2− a␬^1/2

2

冊

^.

Using these expressions, the conservation equation共33兲 be- comes

5

冑

_␲

6 d dt

a

␬⁼

6a^p+1 2共p+ 1兲^3/2

冑

␬⁻

V 2

冑

a␬^−a

冑

_␬. 共35兲 The system of equations共29兲, 共31兲, and 共35兲governing the evolution of the pulse is a closed system of equations for the pulse amplitudea, the pulse width␬⁻¹, and the velocity

␨

˙=V, which together describe its evolution. It is to be noted that when the velocityVin共27兲obtained by the variational method is substituted into the mass conservation equation 共35兲, the fixed point of this equation isa= 0. This nonphysi- cal situation arises from the fact that the Gaussian trial func- tion共10兲is not an exact solution of the ZK equation and thus the conservation equations and variational approximation need not give the same solution, as would be the case if the trial function was based on an exact solution of the govern- ing equations. This difference is shown in Figs. 5共a兲 and 5共b兲. The same situation is encountered in other problems for which an approximate trial function is used to obtain ap- proximate evolution equations. To overcome this nonphysi-

cal result, the velocity is now calculated by considering the evolution of the moment of momentum. This calculation is performed using the same approximations as were made in the derivation of the conservation equations, such as the mass equation共35兲.

From the ZK equation 共8兲it can be shown that the mo- ment of momentum equation is

d

dt

冕

_−⬁^⬁

冕

_−⬁^{⬁ 12xu2dxdy= −}

冕

_−⬁^⬁

冕

_−⬁^⬁

冋

^x⳵⳵x

^冉

^{p6+ 2up+2}

+uuxx−1 2u_x²−1

2u_y²

冊

+ ⳵

⳵^y共uuxy兲

册

^{dxdy. 共36兲}

Using the trial function共10兲and recalling thatu₀u₁= 0, that u₀and u₁ and their derivatives are zero at infinity, and that 兩u₁兩Ⰶu₀, we obtain

d

dt

冉

^␨a␬²

冊

⁼共p²⁴+ 2兲² a^p+2

␬ ^{− 4a2. 共37兲}

Using the momentum equation 共31兲, this gives to leading order

␨^{˙=V= 24a}

p

共p+ 2兲²− 4␬^, 共38兲

which specializes in the casep= 1 to

␨^˙=V=83a− 4␬. 共39兲 Therefore the closed system of approximate equations for the amplitudea, width␬⁻¹, and velocity␨^{˙=Vis now}

d dt

a²

␬ ^{= 0 or} a²

␬ ⁼ a₀²

␬0

, 共40兲

5

冑

_␲

6 d dt

a

␬⁼

6a^p+1 2共p+ 1兲^3/2

冑

␬⁻

V 2

冑

a␬^−a

冑

␬, 共41兲

␨^{˙=V= 24a}

p

共p+ 2兲²− 4␬. 共42兲

They combine to give the single equation for the amplitude 5

冑

_␲

6 da

dt =␬01/2

a₀

冋

^{␴共p兲ap−␬a0}02a²

册

^{a2, 共43兲}

wherea共0兲=a₀,␬共0兲=␬0, and

␴共p兲= 12

共p+ 2兲²− 3

共p+ 1兲^3/2. 共44兲

Forp= 1 there is a stable fixed point given by ae=␴共1兲a₀²

␬0

=8

冑

2 − 9 6

冑

2

a₀²

␬0

. 共45兲

It is important to note that the equilibrium amplitudeaewill be attained in different ways according to the relationship between the initial amplitude and width. If 0⬍␬0⬍␴共1兲a0,

FIG. 3. RegionR共t兲outside caustic.

the initial Gaussian u₀ will absorb mass until it reaches a lump with amplitudeae; if␬0⬎␴共1兲a0, the initial Gaussian will lose mass until it stabilizes to a Gaussian of amplitude ae. The special case ␬0=␴共1兲a0for the initial Gaussian will show very small changes with time.

For other values of p, it is convenient to write the am- plitude equation共43兲in the form

5

冑

_␲

6 da

dt =␬₀^1/2

a₀

冋

^{␴共p兲−␬a}02⁰a^2−p

册

^{ap+2, 0⬍p⬍2, 共46兲}

5

冑

_␲

6 da

dt =␬01/2

a₀

冋

^{␴共p兲ap−2−␬a}02⁰

册

^{a4, 2艋p. 共47兲}

The physical fixed point for the amplitude equation共46兲is

ae=

冉

␬0␴^a02共p兲

冊

^{1/共2−p兲 共48兲}

and a similar time evolution behavior as the one described above forp= 1 occurs. Nevertheless, for the amplitude equa- tion 共47兲 for p艌2 a more careful analysis must be made.

Since ␴共p兲⬍0 for p⬎10.44, the amplitude equation 共47兲 predicts a final lump solution with amplitude

a_e=

冋

^a0^␬0

2␴共p兲

册

^{1/共p−2兲 共49兲}

for 2⬍p⬍10.44, but whenp⬎10.44 the steady lump ceases to exist. Also for 0⬍p⬍2 the fixed point is stable, while for 2⬍p⬍10.44 it is unstable.

For the case p= 2, the only fixed point is a= 0 and the exact solution of the approximate equation共47兲for the pulse amplitude is

a共t兲= a₀

关1 −共18/5

冑

_␲兲a₀²␬₀^1/2共␴共2兲−␬0/a₀²兲t兴^1/3. 共50兲 If the initial conditions for the width and amplitude satisfy

␬0⬍␴共2兲a₀², then the initial Gaussian reaches infinite ampli- tude at time

t^*= 5

冑

_␲

18a₀²␬01/2关␴共2兲−␬0/a₀²兴. 共51兲 On the other hand, if␬0⬎␴共2兲a₀², then the initial Gaussian decays into radiation. This unstable behavior for p艌2 will also be studied numerically in Sec. VI. Similar behavior for the mKdV equation was observed in Ref. 8 and studied using the present approximate method in Ref. 9. In the threshold case␬0=␴共2兲a₀², the approximate equations predict that this particular initial Gaussian 共10兲 will change by a small amount. These results are of particular interest since in Ref. 8 the collision of two solitary waves traveling in the same direction for the casep= 2 was studied. It was found that one of the waves blows up in finite time after interaction. In the present model we have finite time blow up for a family of initial Gaussians, independent of any interaction. The results of Ref. 8 can then be qualitatively explained by noting that the interaction increases the mass of the leading soliton above the threshold described above.

The solution of the amplitude approximate equation共43兲 forp= 1 was compared with a full numerical solution of the ZK equation共1兲for the same initial conditions as in Fig. 1.

The agreement was found to be only qualitative, as expected from previous studies of the KdV equation¹¹and the mKdV equation⁹ where similar behavior was found. To overcome this a more accurate mass and momentum balance will now be analyzed in order to obtain approximate equations which include the effect of the shed dispersive radiation.

However, before considering a more detailed balance of mass and momentum, let us examine the expression共42兲for the velocity. Previous work on the KdV,¹¹mKdV,⁹KP,¹⁰and BO¹⁶ equations shows that the velocity expression arising from the moment of momentum equation is the same as that of the variational formulation. However, it was further found that this velocity expression needed to be slightly modified in order to obtain the correct, numerical rate of decay to the fixed point. In this previous work it was argued that the correct velocity had the same functional form and fixed point as the one obtained by the approximate theory, but the coef- ficients had to be modified in order to give the correct times- cale. For the cases of the BO and KP equations, the range of the coefficients was restricted so that the velocity was nons- ingular. For the KdV and mKdV equations, the single free parameter in the velocity expression was obtained by match- ing to a specific numerical solution and excellent agreement with full numerical solutions was then obtained for a very wide range of initial values. This procedure was suggested by analogy with other mechanical problems. We note that the velocity is an angle variable. We recall from the theory of mechanical systems that an angle variable evolving in a sys- tem with a mean level, such as an oscillator with a quadratic nonlinearity, will have its evolution coupled to the mean level. The final form for the velocity will then have the func- tional form appropriate for the self-interaction, but with co- efficients modified by the mean level. What is not clear is how to compute the localized mean level which couples to the velocity. However, we shall assume the same result valid for simple systems and verify its validity by comparison with full numerical solutions. It is in our view a very interesting problem of how to develop the appropriate perturbation theory to account for this effect. It is noted that in the present case the velocity expression共42兲, which gives the nontrivial fixed point, is the one modified.

In order to preserve the fixed point of the approximate equation共41兲, expression共42兲for the velocityVis modified as

V=␮a^p− 4␬+

冋

^{共p24+ 2兲2−␮}

册

␴^␬共p兲, 共52兲 where␴共p兲was defined above as

␴共p兲= 12

共p+ 2兲²− 3

共p+ 1兲^3/2. 共53兲

This specializes in the casep= 1 to

V=␮a− 4␬+

冉

^83−␮

冊

␴^␬共1兲. 共54兲 A vital point is that the fixed point of the approximate equa- tions does not depend on the value of ␮. These velocity expressions reduce to the fixed point value共39兲when aand

␬ satisfy the fixed point relation ␬⁼␴共p兲a^p, independent of the value of␮^.

To obtain a more detailed momentum balance we see that the radiation shed by the evolving lump transports mo- mentum away from it. The momentum conservation equation 共31兲 does not account for this loss. To include this loss of momentum by the lump, let us consider momentum balance in the region ⌫共t兲 which lies to the right of the curveC of Fig. 3 关including the region R共t兲兴. Then from the modified ZK equation共8兲it can be found that

d dt

冕

_⌫共t兲

1

2u²dxdy=

冕

C共t兲

冉

p^up+2+ 2+uuxx−1 2u_x²

− 1

2u²_y,uuxy

冊

^·ndS−V

冕

C共t兲

1 2u²dS.

共55兲 Here n is the outward unit normal to C共t兲. To derive this momentum balance equation it has been assumed that the curveC共t兲moves rigidly with the velocityV of the lump.

It can be seen from Fig. 1 that as the lump evolves, a flat shelf, of small amplitude relative to the lump, forms, which is attached behind the lump. Let the amplitude of this flat shelf beu_⬁. Since the shelf is flat and the derivative terms in the momentum balance equation共55兲are quadratic, they can be neglected. Evaluating the integral on the left-hand side of 共55兲using the trial function of共10兲, we then have

d dt

␲ 4

a²

␬ ^{= −}

冕

C共t兲

冉

^{pu⬁p+2+ 2+ 12Vu⬁2}

冊

^{dS. 共56兲}

This momentum equation gives the momentum lost from the lump to the shed dispersive radiation. Finally, if it is assumed that the length of the curveCfor which the shelf is flat isᐉ and thatp is larger than 1, the momentum balance equation for the lump is

d dt

␲^a2

␬ ^{= − 2Vᐉu⬁2. 共57兲}

To conclude the momentum balance equation, we need to determine the amplitudeu_⬁ of the shelf by relating it to the mass lost by the lump. To this end, we reconsider mass balance, including the trailing flat shelf. Now the shelf is bounded by a caustic. Therefore, as for momentum balance, it can be shown from the modified ZK equation共8兲that the mass balance equation is

d dt

冕

_⌫共t兲

u dxdy= −

冕

C共t兲

冉

p^up+1+ 1+uxx+uyy

冊

^dS

−V

冕

C共t兲

u dy. 共58兲

To derive this mass balance expression it has again been

assumed that the shelf is rigidly attached to the lump and moves with velocityV.

As above, we now take the shelf to be flat and to have small amplitude relative to the lump, as is illustrated in Fig.

1共b兲, and take u_⬁ to be the amplitude of the shelf at C.

Therefore, as for momentum conservation, the derivatives in 共58兲can be neglected. Then calculating the mass of the lump from the trial function in共10兲, we have

d dt

␲^a

␬ ^{= −}

冕

C

冉

^{pu⬁p+1+ 1+Vu⬁}

冊

^{dS. 共59兲}

The integral on the right-hand side of this mass balance equation is again approximated by taking the length of the curveC for which the shelf is flat to be ᐉ, so that the final mass balance equation for the evolution of the lump is

d dt

␲^a

␬ ^+ᐉu⬁

冉

^{V+ pu+ 1⬁p}

冊

^{= 0. 共60兲}

This equation determines the mass of the lump, incorporating the mass lost from the lump to the shed dispersive radiation.

Since we are interested in the case for which p艌1 and u_⬁ Ⰶ1, we have

u_⬁= − 1 ᐉV

d dt

␲^a

␬ ^{, 共61兲}

which is the same expression for the shelf amplitude as for the KP equation.¹⁰ Using this expression for u_⬁ in the mo- mentum balance equation 共57兲 gives the final momentum equation for the lump as

d dt

␲^a2

␬ ⁺ 2

ᐉV

冉

^dtd␲␬^a

冊

^{2= 0. 共62兲}

This momentum equation, together with the mass equation 共35兲, is the system of approximate equations for the evolu- tion of the lump.

The system of approximate equations共35兲for mass,共52兲 for the velocity, and共62兲 for momentum form a closed sys- tem of equations for the lump amplitude a and width ␬^−1, which are the collective coordinates. It is noted that the fam- ily of fixed points is independent of ᐉ and ␮. Solutions of this system of approximate equations will be compared with full numerical solutions of the modified ZK equation in Sec.

VI. Before these numerical and approximate results are com- pared, the approximate equations for the three-dimensional case will be derived in the next section.

IV. EXTENSION TO THREE SPACE DIMENSIONS Due to the simplicity of the caustic and the trial function for the ZK equation, it is possible to extend the calculation of the approximate equations to the three space dimensional version of the ZK equation. To this end, the three- dimensional trial function

u₀=a共t兲e^{−␬关共x−␨共t兲兲2+y2+z2兴} 共63兲 is used for the three-dimensional ZK equation 共4兲. Exactly the same procedure as for the two-dimensional ZK equation can be carried out, based on equations for conservation of

mass and momentum. The caustic is now a surface, but since this caustic surface is conical and the trial function is Gauss- ian, the integrals involved in averaging the conservation equations can be carried out in detail.

With these extensions to three space dimensions, the ap- proximate equations for the evolution of a lump for the three-dimensional ZK equation 共4兲 are, as before, obtained from the equations for conservation of mass and momentum and the equation for moment of momentum in the form

冑

_␲

冉

^{1 +}

^冑

²³

冊

^dtd␬^a3/2= 1 2共1 +p兲²

a^1+p

␬ ⁻ Va

2␬^{−a, 共64兲} d

dt a²

␬^{3/2= 0, 共65兲}

V= 4

冑

2

共2 +p兲^5/2a^p− 5␬^, 共66兲

when the loss of momentum to the shed radiation is ne- glected. As before,V denotes the velocity. It is possible to reduce these approximate equations to a single equation for the pulse amplitudea in terms of the initial values a₀for a and␬0for ␬. This equation is

da dt =

冑

_␬0

a₀^3/2

冉

^{␥共p兲−␤a␬}04/3⁰ a^4/3−p

冊

^{ap+5/3, 0艋p⬍43, 共67兲}

da dt =

冑

␬0

a₀^3/2

冉

^{␥共p兲ap−4/3−␤a␬}04/3⁰

冊

^{a3, 43艋p, 共68兲}

where the function␥ ^is

␥共p兲=␲^−1/2

冉

^{1 +}

^冑

²³

冊

⁻¹

冋

^{共2 +2}

^冑

^{p兲25/2−2共1 +1 p兲2}

册

^{, 共69兲}

and

␤^{= 3}

2

冑

_␲

冉

^{1 +}

^冑

²³

冊

^{−1. 共70兲}

A straightforward analysis of these equations shows that for p⬍⁴₃ the lump soliton fixed point is stable and is ap- proached asymptotically. On the other hand, for ⁴₃艋p 艋25.61 the soliton fixed point

ae=

冉

^{␥␤共p兲a␬}04/3⁰

冊

^{1/共p−4/3兲 共71兲}

acts as a threshold. Lumps with initial amplitudes below this threshold decay into radiation and lumps with initial ampli- tudes above the threshold narrow and collapse in finite time.

Finally, forp⬎25.61, the soliton fixed point ceases to exist and all initial conditions decay into radiation. This behavior has not been verified numerically due to the amount of cal- culation required in three space dimensions and is beyond the scope of the present work. Finally in this section, the argument used to include the effects of shed dispersive ra- diation in the momentum equation will now be outlined.

Let us consider the mass conservation equation for the total of the mass of the lump and the massM₁of the radia- tion

␲^3/2d dt

a

␬^3/2+ dM₁

dt = 0. 共72兲

As in the two-dimensional case of the previous section, it is assumed that a shelf of cross-sectional areaAand of approxi- mately constant amplitudeu=u_⬁is formed inside the caustic surface which travels at the same velocity V of the lump.

Assuming 兩u_⬁兩Ⰶ1, then the approximate equations for the three-dimensional pulse can be derived in the same manner as for the two-dimensional equations共33兲–共35兲. In particular, as for the two-dimensional mass equation共59兲, the derivative of the radiation massM₁can be approximated by

dM₁ dt = 6A

p+ 1u_⬁^p+1+VAu_⬁. 共73兲 We thus obtain

␲^3/2d dt

a

␬^3/2+ 6A

p+ 1u_⬁^p+1+VAu_⬁= 0. 共74兲 The conservation of momentum equation, including momen- tum loss to radiation, is obtained in a similar manner as for the two-dimensional case of the previous section, withAand u_⬁as for mass conservation,

␲^3/2 4

冑

2 d dt

a²

␬^3/2+ 6A

p+ 2u_⬁^p+2+1

2AVu_⬁²= 0. 共75兲 The same velocity expression 共66兲 obtained on neglect of momentum loss to radiation is now used for the velocity V, since momentum loss to radiation is small.¹¹The resulting equations for the evolution of the three-dimensional pulse are then

␲^3/2d dt

a

␬^3/2+ 6A

p+ 1u_⬁^p+1+VAu_⬁= 0, 共76兲

␲^3/2 4

冑

2 d dt

a²

␬^3/2+ 6A

p+ 2u_⬁^p+2+1

2AVu_⬁²= 0, 共77兲

V= 4

冑

2

共p+ 2兲^5/2a^p− 5␬^. 共78兲

V. APPROXIMATE EQUATIONS FOR SURFACE ELECTROMIGRATION

The derivation of the approximate equations for surface electromigration, governed by the ZK equation共2兲and Pois- son’s equation共3兲, is very similar to that for the modified ZK equation of the previous section. Therefore, only the main changes from the derivation of the previous section will be discussed in detail. As was remarked at the end of Sec. II, comparing Figs. 1 and 2 shows that lump evolution for the ZK equation共1兲is very similar to that for the coupled system 共2兲 and共3兲for electromigration. The only change is due to the convective termⵜu·ⵜ⌿. The system 共2兲and 共3兲 again