Numerical solutions of the weakly and fully non-linear ODEs

we initially set the number of spectral terms N = 61, and the number of radial pointsP = 1000, with the maximum radius set toR = 25. Setting the initial guess Gaussian function parameters in (6.54) to B = 2, µ = 0, and σ = 4.1, fsolve

with default function tolerance (‘TolFun’=1e-6) converged to the solution shown in Figure 6.1.

Using exactly the same parameters, but with the number of spectral terms increased to N = 101, we obtain the same solution as in the N = 61 case, with negligible difference. Setting N = 60 and N = 100, keeping everything else the same, the residuals converge to slightly different distributions of similar size. Using the func- tion obtained from theN = 61 case as a guide, we make a fresh initial guess with the new Gaussian parametersB = 0.7,µ= 6 and σ= 1.5, with‘TolFun=1e-12’. This initial guess turned out to be much more accurate, converging to the solution shown in Figure 6.2, but stalling due to the minimum function tolerance being reached. The same routine was run with N = 101, which yielded the same solution. For the N = 101 case, the residuals reduced to<±5×10−6 from values on the order of 104 based on the initial Gaussian guess in Figure 6.2. The values of the right hand side of the implicit ODE itself (6.50) were reduced from an order of 104 to<±0.2, where ideally they should be zero at all radial points. Such a dramatic relative difference in the Galerkin residuals and implicit ODE values between the two plots in Figure 6.2 strongly suggests the presence of a real solution for (6.50), or at least an extreme local minimum in the solution phase space.

With a solution to the weakly non-linear ODE (6.50) in hand, we turn now to finding solutions to the fully non-linear system (6.31), which is the same as the weaker equation, except that it lacks the square root term with ambiguous sign. At this point, we assume that the J(χ) solution in Figure 6.2 provides a good initial guess to finding solutions of the full ODE. Our initial guess takes the form

j_a=J. (6.57)

Setting to a “small” value of 10−2, we find that the initial guess J immediately satisfiesfsolvewithout any further iterations. Therefore for small, (6.57) appears to be a good approximate solution to (6.31). This result is not unexpected, since the square root term in (6.31) is negligibly small in the limit of small . Interestingly, it turns out that if we choose = 1, which is not “small”, so that our initial guess is j_a,init = J, we still obtain a solution for the fully non-linear ODE system, after many iterations. The solutions for both sign cases obtained using the parameters N = 101, P = 1000, = 1, and ‘TolFun’=1e-12 are displayed in Figure 6.3. Comparing these solutions with the J(χ) solution in Figure 6.2, we can see that the initial and final guesses for j_a closely match, despite the fact that the residual values have been reduced from ∼ 104 to ∼ 10−4, and the ODE values (this time corresponding to the fully non-linear case) have been reduced from ∼ 104 to ∼ 2 through the action of fsolve. Now, due to the presence of the square root term in (6.31), a small imaginary part on the order of∼10−4to∼10−3times the magnitude of the real part is present in the final forms forj_a,j_a,χ, etc. We are assuming that

6.1. STATIC, SPHERICALLY SYMMETRIC REDUCTION 75

all imaginary components are erroneous artefacts produced by thefsolvealgorithm when dealing with the square root term, and are therefore ignored.

The solutions displayed in Figure 6.3 appear to be static “soliton-like” solutions to the Maxwell-Dirac equations under spherical symmetry. These solutions are of course, not technically solitons since they do not evolve with time, but radially they closely approximate a characteristicsech2(χ) soliton profile [47], and have the feature of being localized due to the non-linearity of the system self-interaction. Physically, these solutions correspond to a static, hollow shell of electric charge. 6.1.8 Numerical solutions using multiple Gaussian initial guess In principle, we can have as many Gaussian functions in our initial guess for J(χ) as we want, which is useful when searching for solutions with multiple peaks. The corresponding generalizations of (6.54)-(6.56) are

Jinit(χ) = M X m=1 Bmexp −(χ−µm) 2 2σm2 , (6.58) b0 = 1 R Z R 0 M X m=1 Bmexp −(χ−µm) 2 2σm2 dχ, (6.59) bn= 2 R Z R 0 M X m=1 Bmexp −(χ−µm) 2 2σm2 cos(αnχ)dχ. (6.60) Considering the case whereM = 2, we now have six parameters to choose from when setting Jinit. After some experimentation, we find that choosing the parameters B1 = B2 = 0.23, µ1 = 5.0, µ2 = 8.4, σ1 = σ2 = 0.8, with ‘TolFun’ and ‘TolX’ both set to 1e-7, yields the weakly non-linear solution shown in Figure 6.4. The solution curve corresponds to a reduction of the residuals from ∼ 102 _{to ∼ 10}−6_, and a reduction of the magnitude of the ODE values (as a function ofχ) from∼102 to ∼ 10−2, compared with the initial double-Gaussian curve. Bootstrapping this weakly non-linear J(χ) solution by using it as the initial guess for the fully non- linear scheme, yieldsj_a(χ) solutions for both positive and negative square root sign cases, with negligible difference to theJ(χ) form in Figure 6.4.

A comparative plot of single and double humpj_a(χ) distributions is given in Figure 6.5. From this, an obvious question arises: does there exist some set of solution eigenvectorsj_a,n(χ), whereby the number of peaks increases as the ordernincreases? It is interesting to speculate in this manner, however the analytical extraction of such a set of eigenvectors from (6.31) or (6.50), if any actually exist, would be quite a formidable task indeed. We have demonstrated numerically that solutions of multiple “orders” do exist, but there are serious efficiency problems associated with our simple calculational scheme. Searching for multiple hump solutions with M >1 is quite computationally intensive, with calculations often having to run for several days on end. It may be the case that using a set of basis functions that are more suited to the symmetry of the problem, such as Laguerre or Chebyshev polynomials, will improve efficiency, as well as providing greater insight into the nature of the problem.

6.1. STATIC, SPHERICALLY SYMMETRIC REDUCTION 76

Figure 6.1: J(χ) solution, given parameters N = 61, B = 2, µ= 0, σ = 4.1, and R = 25. Default tolerance. Dotted line is the initial Gaussian guess, solid line is the solution function.

Figure 6.2: J(χ) solution, given parameters N = 61, B = 0.7, µ= 6,σ = 1.5, and R= 25. Function tolerance of 10−12. Dotted line is the initial Gaussian guess, solid line is the solution function.

6.1. STATIC, SPHERICALLY SYMMETRIC REDUCTION 77

Figure 6.3: Dimensionless j_a(χ) solutions to the static, spherically symmetric Maxwell-Dirac equation ODE, where the solid and dotted lines correspond to posi- tive and negative signs on the square root term respectively.

Figure 6.4: J(χ) solution, given parameters N = 101, B1 = B2 = 0.23, µ1 = 5.0, µ2 = 8.4, σ1 = σ2 = 0.8 and R = 25. Function and step size tolerance of 10−7. Dotted line is the initial M = 2 double-Gaussian guess, solid line is the solution function forJ(χ), and a close approximation of the fullj_a(χ) solution.