One major fascination of solitons is their character simultaneously as waves and particles. Solitons collide, interact, and emerge unchanged except for a phase shift. The analogy to subatomic particles is compelling. But more interesting than solitons is the behavior of the solitary solutions to the Regularized Long-Wave (RLW) equation in which solitary solutions collide and may create secondary solitary waves or sinusoidal solutions. This corresponds even better to subatomic phenomena in that collisions of particles create other particles and/or radiation. The purpose of this paper is to study the process for the creation of the secondary waves. This provides the opportunity to investigate the creation process for possible insights into the corresponding processes of particle physics.

II. The Regularized Long-Wave Equation

The RLW equation has the form

u_t + u_x + uu_x - u_xxt = 0

whereas the KdV equation may be expressed as

u_t + u_x + uu_x + u_xxx = 0.

The RLW equation was formulated by Peregrine (1966) as an alternative to KdV equation for studying soliton phenomenon. It was proposed because it would not have the same limitations for the size of the time step in numerical solution that the KdV has.

Traveling Wave Solutions

Some aspects of the solutions to the KdV and RLW equations may be derived from analysis. A traveling wave solution is of the form u(x-vt-x₀). Letting z=x-vt-x₀ the RLWE becomes:

-vu_z + u_z - ½(u²)_z - vu_zzz = 0

Integrating this once with respect to z gives

−vu + u + ½u² - u_zz = c₀

where c₀ is an arbritrary constant.

Multiplying the above equation by u_z gives and equation that can be integrated with respect to z to give

−½vu + u + (1/6)u³ - u_zz = c₀

Finally the equation can be solved for u_z and integrated. It is readily found that the RLWE has solitary wave solutions of the form a·sech²(b(x-vt-x₀). That is to say, if at t=0, u=a·sech²(bx) then the solution for all future times will be of that form. This is a form found frequently in soliton solutions. When v is positive the waveform moves to the right and when it is negative the wave form moves to the left.

Collisions of solitary solutions are created by taking initial conditions which are the sum of two or more solutions. The collision occurs at the value of x and t such that the argument (x-vt-x₀) is zero for both waves. It is shown elsewhere that the parameters a, b, and v must satisfy the following conditions:

v=1/(1-4b²)
and
a=12b²/(1-4b²).

These conditions are equivalent to

b = ½[a/(a+3)]^½
and
v = l+a/3.

It is easily seen that for b to be real a must be greater than zero or less than -3. This forbidden interval of (-3,0) for a is of importance. It means that v will be greater than 1 or less than 0. Note that the larger the magnitude of the amplitude the closer the value of the parameter b is to (1/2) and consequently the larger the speed of the wave. The parameter b indicates how sharp of a peak the wave has. The width of the wave, as measured by a e-ratio or standard deviation, is inversely proportional to b. Thus the larger the magnitude of the amplitude the narrower the wave. The minimum width is 2.

Solitary Waves Versus Solitons

Initially it was thought that because of the similarity of the RLW equation to the KdV equation that it would also have n-soliton solutions. Eilbeck and McGuire found in 1975 that after the collision of two solitary waves the same waveforms reappeared with amplitudes within 3/10 of 1 percent of the originals. It was concluded that the RLW equation had soliton solutions, but Abdulloev (1976) found that the discrepancy could not be accounted for by numerical inaccuracies.

Santarelli in 1978 found that secondary solitary waves are created by the collision of large amplitude solitary waves. The results of Santarelli's study are shown in Figures 1 through 9.

In Figure 1 the positive amplitude wave is to the left of the negative amplitude wave.

In Figure 2 the waves are colliding.

In Figure 3 the waves are superimposed but higher magnitude of the negative amplitude wave results in it predominating. There is a slight change of scale for Figure 3 to allow the display of more detail. It must be emphasize that these results are not simply from the addition of the two initial waves. The two initial waves are used as a initial conditions for the numerical solution of the RWL equation.

In Figure 4 the waves have effectively passed through each other, but with diminished amplitude.

In Figure 5 the secondary waves can be seen developing. There is a notch on the negative amplitude wave that will develop into the secondary negative amplitude wave. The secondary positive amplitude is also developing but it is less obvious at this stage.

In Figures 6 through 9 the secondary waves have developed and continue to move; the positive amplitude to the right and the negative amplitude waves to the left.

Lewis and Tjon (1979) using even larger amplitude solitary waves found that multiple secondary waves are created, always in pairs with positive and negative amplitudes. Furthermore, Lewis and Tjon found resonance effects in which the relationships between the parameters of the solitary waves became important. Figure 2 shows some of Lewis and Tjon's results. They also found cases in which the the positive and negative solitary waves annihilate each other and create a sinusoidal residual solutions called appropriately radiation.

III. The Invariants of the RLW Equation

According to Lewis and Tjon the known conserved quantities of the RLW equation are:

∫udx, ∫(u² + u_x²)dx, and ∫(1/3)u⊃ - u_x²Jdx,

where all of the integrations are over the interval [−∞,+∞]. For u(x) =a·sech²(bz) where z=x-vt-x₀

∫udx= (a/b)∫sech²xdx

so the first invariant is equivalent to (a_i/b_i) for the waves before the collision being equal to (a_j/b_j after the collision. The second invariant for a solitary wave a·sech²(bz) is equal to

a²∫sech⁴(bz)dx + 4(ab)² sech⁴(bz)tanh²(bz)dz

which in turn is equal to

[a²(1-4b²)/b]∫sech⁴ ydy - a²b·sech⁶ydy.

Since ∫sech⁶ydy/∫sech⁴ydy = 4/5 the second invariant is proportional to a²[(1/b)+(16/5)b] Because the integrand of the second invariant is nonlinear it cannot be asserted that its value for u₁+u₂ is the sum of its values for u₁ and u₂.

However if the waves for u₁ and u₂ are widely separated so that u₁u₂ is virtually zero then additivity is closely approximated. Thus a²[(1/b_i)+(16/5)b_i] long before the collision is equal to its value long after the collision.

The third invariant for a solitary wave is proportional to a²[(4a/15b) + (4/5)b] and so the sum of these quantities before the collision is equal to the sum for the waves long after the collision.

Table 1 gives the parameters of the incident waves in the collision studies and the secondary waves. Unfortunately the studies do not give the parameters of the primary waves after the collision so one cannot use the results to illustrate the invariances. The changes in the primary waves were small so they were neglected.

It is possible to make some interesting computations from the results of the studies. In both studies the magnitude of a/b for the left-moving secondary wave is 46 percent of the combined magnitude of a/b for both secondary waves. The value of the second invariant for the negative amplitude wave is 64.01 whereas its value for the positive amplitude secondary wave is 5.44. Thus there is no equipartition principle involved in the creation of the secondary solitary waves. The values of the third invariant for the negative and positive amplitude secondary waves are -1.25 and 1.53, respectively.

A collision of two primary waves in which two secondary waves are produced involves four unknown parameters, the amplitudes of the primary waves after collision and the amplitudes of the secondary waves. There are three invariances but they are nonlinear so it is not possible to say whether the unknown parameters are determined uniquely by the invariances; i.e., the conservation laws. Actually the after-collision waves cannot be a unique solution of the invariances because the there is always the solution of the after-collision waves being equal to the before-collision waves.

IV. Analysis of the Secondary Wave Creation

Let u₁ and u₂ be two solutions to the RLW equation and let u₃ be a solution with the initial conditions equal to u₁+u₂ at t=0.

The RLW equation may then be put into the form

(u−u_xx)_t+(u+½u²)_x = 0.

Now let w=u₁−u₂. Since u₁ and u₁, satisfy the RLW equation it follows that w satisfies the equation

(w-w_xx)_t +(w+½w²)_x+(wu₁)_x+ (wu₂)_x+(u₁u₂)_x = 0.
 

This is the equation which describes the formation of secondary solitary waves. The initial condition is w(x)=0 for all x. A collision in where (u₁u₂) is nonzero and such that (u₁u₂)_x is nonzero. This quantity, the gradient of (u₁u₂), is what drives the creation of the secondary waves. From the numerical studies it is reasonable to conclude that w will consist of two separated parts; one associated with the positive amplitude solitary wave and one with the negative amplitude wave.

A rearrangement puts the equation into the form

(w_xx-w)_t = (w+½w²)_x+(wu₁)_x+ (wu₂)_x + (u₁u₂)_x = 0.
 

A form for numerical solution is

(Δw_xx-Δw) = Δt [(w+½w²)_x+(wu ₁)_x+ (wu₂)_x+(u₁u₂)_x]
 

where Δw is w(t+Δt)-w(t).

This is an equation of the form

y_xx − y = f(x)

which has the solution

y(x) = ∫_−∞^xsinh(x-z)f(z)dz

Therefore the numerical solution for the secondary wave amplitude w is given by

w(x,t+Δt) = w(x,t) + Δt∫_−∞^xsinh(x-z)f(z)dz
where f(x) is given by
f(x) = (w+½w²)_x+(wu₁)_x+ (wu₂)_x+(u₁u₂)_x
or, in a more computationally efficient form
f(x) = [w((1+½w)+u₁+ u₂))+(u₁u₂)]_x
 

Since the forcing term f(x) is a derivative it is best to make use of integration by parts to eliminate the need to approximate the derivative. Thus let f(x)=F_x(x) then

∫_−∞^xsinh(x-z)f(z)dz = [sinh(x-z)F(z)]_−∞^x − ∫_−∞^xcosh(x-z)F(z)dz

The limit of sinh(x-z)F(z) as z→−∞ needs to be investigated but for now it is presumed to be zero. Since sinh(0)=0, the estimating equation for computation is

w(x,t+Δt) = w(x,t) − Δt∫_−∞^xcosh(x-z)F(z)dz
where F(x) is given by
F(x) = (w+½w²)+(wu₁)+ (wu₂)+(u₁u₂)
or in the computationally more efficient form
F(x) = w(1 + ½w + u₁ + u₂) + u₁*u₂
 
 

Suppose that u₁ and u₁ are nearly Dirac delta functions of opposite signs that coincide (except for sign) at x=0 at t=0. Initially (t=0) the F function will be zero everywhere except at x=0 where u₁ and u₂ coincide. At the first time step Δt, then w(x,Δt) will be -Δt*cosh(x)∫F(z)dz for x>0,

where ∫F(z)dz denotes the integral across the interval about zero. For x<0 the convolution integral will be zero because z≤x and thus the integration never covers the interval for which F(z) is nonzero.

The w function will be zero for x<0 and some positive value, say w₀ for x>0. For t≥0 the product u₁u₂ will be zero everywhere.

Before dealing with the interaction of two solitary waves it is worthwhile to show how such waves procede without interaction, as is shown below. In this diagram the images of past positions are not erased.

So the initial time step Δt produces values for the w function of the form

w(x,Δt) = 0 for x<0
= -Δt[cosh(x)A] for x≥0

where A=∫F(z)dz.

The value of Δt should be on the order of the time it takes for u₁ and ₂ to slip past each other. This would be sech² waves on the order of (1/b₁+1/b₂)/(|v₁|+|v₂|). For the next time step u₁u₂ is zero and the w function is determined by the term w*(u₁+u₂), so except for the intervals covered by u₁ and u₂ there will not be any augmentation of the level of w.

Once the level of w over some interval is such that the w+½w² term is significant then the level of w in that interval grows on its own without any help from the u₁ or u₂ functions.

(To be continued.)

References: