Solitary waves and solitons are briefly discussed in their historical context, especially regarding the motivation to use solitons for particle models in the nonlinear Klein-Gordon (NKG) equation. Conservation equations for charge, energy, and momentum follow from Noether’s theorem in the usual way, and the NKG equation can be coupled to Maxwell’s equations in the standard, gauge invariant manner. A recently proposed model is summarized in which two NKG equations are coupled to the electromagnetic field. In that model, solitons mimic the dynamical behavior of electrons and protons. A new result is then presented that follows from that model. Although the model is purely classical, it turns out that the arc spectrum of hydrogen is emitted into the electromagnetic field when small oscillations of one Klein-Gordon equation occur in the vicinity of a proton-like soliton. It is perhaps unexpected that a purely classical model can exhibit behavior suggestive of a phenomenon that is generally presumed to occur only in a quantum-mechanical context. Because of the way in which that result occurs, however, it is not clear whether there is any possible relevance to the actual physical phenomenon.
- nonlinear systems and models
- nonlinear Klein-Gordon-Maxwell equations
- particle models
- hydrogen spectrum
- nonlinear field theories
When a term that acts as a restoring force is added to the usual linear wave equation, the equation
is obtained. Schweber (, p. 54) points out “When Schrödinger wrote down the nonrelativistic equation now bearing his name, he also formulated the corresponding relativistic equation.” Within the year of 1926, Klein, Gordon, and at least three others independently proposed the use of Eq. (1) as the relativistic generalization of Schrödinger’s equation. Eq. (1) has since become known as the Klein-Gordon equation. Instead of the original time and space variables that might be used in a quantum-mechanical context, we have chosen to write Eq. (1) using dimensionless variables
where are time and space coordinates in customary units (e.g., seconds and meters). Similarly,
When one attempts to use the Klein-Gordon equation to describe the quantum-mechanical problem of an electron in the Coulomb field of a nucleus of atomic number
The Klein-Gordon equation is, nevertheless, a simple example of a relativistic partial differential equation and, as such, it is second in importance only to the wave equation. The relativistic invariance is preserved even when the
Here, we are interested in the soliton solutions that occur in the NKG equation, so Section 2 is a brief historical overview of solitary waves and solitons, followed by the historical context in Section 3 that motivates the use of solitons in classical field theories. Before coupling the NKG equation to Maxwell’s equations in Section 8, we need to examine the NKG equation itself in more detail in Sections 5 and 6. An explicit example of a soliton solution is given in Section 7. In Sections 9–11, we summarize a recently proposed model in which a second NKG equation is introduced so as to model solitons that can be thought of as protons as well as electrons. In that model, the proton and electron-like solitons attract and repel each other in the desired way. Finally, in Section 12, we present new results from the same model where it is shown that small oscillations in the neighborhood of a proton-like soliton emit the well-known frequencies of the hydrogen spectrum into the electromagnetic field. Since emission of the hydrogen spectrum is a phenomenon that would normally be expected to occur only in a quantum-mechanical setting and not in the present classical field model, we interpret this result in Section 13 and suggest the future direction of the present research.
2. Historical background of solitary waves and solitons
The study of solitary waves has a long history , which originated with observations by Scott Russell (, [6, p. 118]), of an isolated wave moving in a canal. Following along on horseback, he was able to observe that the wave traveled a great distance with little change in form and only gradually died out. The linear theories of water waves available at that time predicted, however, that such waveshapes would necessarily disperse, so for some time Scott Russell’s observations were not taken seriously. The full equations for water waves, even in the irrotational, inviscid case, present quite a formidable boundary value problem, so various simple model equations have been examined in much greater detail. Strangely in contrast to the linear theory, a certain nonlinear model of water waves for shallow water predicted that waves of all shapes would steepen and ultimately break. Eventually, various simple, nonlinear model equations such as the equation of Korteweg and deVries (often known as the KdV equation)
were formulated, where both the dispersion and steepening effects are included. In these equations, the two effects balance each other and do, in fact, allow an isolated waveshape to propagate without change of form.
The term soliton was later introduced to emphasize that solitary waves, like particles, tend to maintain their identity. When a nonlinear term is introduced into the usual linear Schrödinger equation, the nonlinear Schrödinger equation
known (a bit flippantly) as the sine-Gordon equation.
Remarkable methods, such as the inverse scattering method, have been devised for use with certain model equations. With these methods, it is possible to find exact solutions that show solitons colliding and passing through each other while still maintaining their identity. The subject of solitary waves and solitons has been extensively described in books and review articles [9, 10, 11, 12, 13]. For more recent work, see [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and references therein.
3. Use of solitons as particle models
The motivation for the use of solitons as particle models can best be understood by comparison with the historical development of physical theories, especially electromagnetism and quantum mechanics. Despite the successes of Maxwell’s theory of electromagnetism in the late 1800s and the clarified understanding of its transformation properties according to Einstein’s special theory of relativity, classical physics seemed to reach an impasse at the beginning of the twentieth century. The notion of the electron as a point particle was immediately inconsistent with electromagnetism since the electromagnetic energy around a charged point particle is easily calculated to be infinite. Moreover, if an electron were to orbit a proton like a planet around the sun, it would radiate into the electromagnetic field at a frequency based on the orbital speed and quickly spiral inward toward the nucleus.
Early in the twentieth century, however, Planck, Einstein, and Bohr advanced daring new hypotheses to describe blackbody radiation, the photoelectric effect, and the hydrogen atom, respectively. Dirac, Heisenberg, and others later developed various versions of a new quantum mechanics more sophisticated but certainly no more intuitively comprehensible. Quantum mechanics has been perpetually troubled by divergences, as when Bethe  somewhat jokingly referred to a quantity that “…comes out infinite in all existing theories, and has therefore always been ignored.” Remarkable progress was later achieved in quantum electrodynamics by Tomonaga, Schwinger, Feynman, and others, but still only with the aid of a rather arbitrary (some would say procrustean) renormalization procedure. Dirac  concludes his book with the thought that “It would seem that we have followed as far as possible the path of logical development of the ideas of quantum mechanics as they are at present understood. The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage from Bohr's orbit theory to the present quantum mechanics.”
There have been various attempts, notably by de Broglie [7, 26], to gain insight into a possible “drastic change in the foundations of the theory” as envisioned by Dirac, but so far such attempts have had little success. One possible approach, which apparently occurred to many people, is to ask whether solitons could be used to model actual particles as localized regions where the field is large. If point particles were replaced by solitons one could hope for a theory along more classical lines where problems with infinite, divergent integrals would be avoided in a natural way. The line of research described here is meant to be a step, if only a tiny step, in that direction. A recently proposed model is summarized in Sections 9–11. A new result is presented in Section 12, where a phenomenon suggestive of quantum-mechanics occurs, but in an unexpected context.
4. Topological and nontopological solitons
It was found early on that if one tries to use the NKG Eq. (4) to form a localized, particle-like solution, where
Here, however, we will be concerned with another approach to stability that uses the NKG Eq. (4) but with a complex-valued variable
5. Conservation equations for the NKG equation
To understand soliton solutions for the NKG equation, it is important first to see that the NKG equation can be derived from a variational principle. The Lagrangian density is
and the corresponding Euler operator (with respect to
When we set the Euler operator (10) to zero (and take the negative of the complex conjugate), we obtain the NKG Eq. (4). It is well known from Noether’s theorem [4, 35, 36] that each symmetry of the variational principle leads to an equation in what is referred to as conservation form. Once the equation is in conservation form, it is straightforward to integrate in order to show that a certain quantity is conserved. Symmetry with respect to time translation thus leads to conservation of energy, and symmetry with respect to translation in
Since the variational principle that follows from Eq. (9) is unchanged under time translation, it follows that energy is conserved. Specifically, we can rewrite the expression
in the following conservation form:
where the integral is to be taken over a certain region in
Similarly, since the variational principle is invariant under translation in the
is defined as the momentum in the
Next, because the variational principle is invariant when
will be defined as the charge (contained within the region in question), for reasons that will become clear when the NKG equation is coupled to Maxwell’s equations. Again, by the divergence theorem, the charge within the region must remain constant in time except for any charge that enters or leaves through the outer surface.
6. Rotating nonlinear solutions for the NKG equation
Now let us look for solutions of the NKG equation that are rotating in the complex plane with angular frequency
Generally, we will be interested in localized solutions, and
and the energy is
where the integrals are now taken over all of
If we look for a radially symmetric solution
so that Eq. (18) can be rewritten as
7. An explicit soliton solution for the NKG equation
To fix ideas, let us derive the following standard result in somewhat more than the usual detail. An expedient that has been frequently used [37, 38, 39, 40] is to let
There will be a whole family of rotating solutions for Eq. (18), as was mentioned in Section 4. It is convenient to use
If we wish to have
Collecting the above results, we have
for . The result has been written in terms of scaled quantities
Then, we need
The shape of the solution, as given by the dependence of
If a graph of charge
8. Coupling of the NKG equation to the electromagnetic field
Next, we want to couple the NKG equation to electromagnetism to obtain the nonlinear Klein-Gordon-Maxwell (NKGM) system of equations, as has been done by Rosen , Morris , and others. Again it is important to express the coupling of the NKG equation to the electromagnetic field in terms of a variational principle so that the conservation laws of interest can be obtained from Noether’s theorem. Although Maxwell’s equations are usually expressed in terms of the quantities
Here, the notation
It should be noted that
When oscillations of
Then by comparison with Eq. (3), it is clear that a nucleus of atomic number
The Lagrangian density for the electromagnetic field can be expressed in terms of
respectively. Substantial work has been done regarding existence of solitons in the NKGM system [43, 44, 45], in some cases even with nonzero angular momentum . Stability of solitons in the NKGM system is a difficult subject, but some progress has been made [17, 21, 44], especially in the case of small coupling of the NKG equation to the electromagnetic field. The effect of an external field on a soliton has also been examined in the NKGM context by several authors [20, 46, 47].
9. A model with interacting solitons
Recently, a model (call it Model One) has been proposed  in which a second NKG equation
is also coupled to the electromagnetic field, but scaled with the constant
In addition to Eq. (34) for
10. Electron-like solitons
We want to find a soliton solution that satisfies both Eqs. (44) and (45) and can be thought of as an electron. Suppose that we start with a soliton solution for Eq. (18) of the kind that was previously discussed. We want the coupling to the electromagnetic field in Eqs. (44) and (45) to result in only a small change, so that
In the present NKGM context, it turns out Eq. (16) is to be replaced by
so for a solution of the form (43), the charge is now
The divergence theorem together with Eq. (45) shows that for large
As mentioned in Section 4, there is a fundamental difficulty with the use of nontopological solitons as particle models. A whole family of soliton solutions typically exists, and the different solutions will have various values of charge, as was illustrated in the example of Section 7. Thus, it is not yet clear whether a preferred solution occurs in practice that could be regarded as defining one unit of elementary charge. Morris  has made a suggestion in this regard, but it seems that the resolution of the question is beyond the scope of present research on models of this nature since it may well involve complex, possibly chaotic, interactions of many solitons. A second problem is that the stability proofs currently available apply only in the limit of small coupling, so it is not clear whether orbital stability is achieved when the coupling is sufficient to correspond to a meaningful physical case. Future research should resolve this question, although perhaps in a somewhat nonrigorous manner based at least in part on numerical calculation. Despite these two difficulties let us assume for present purposes that a suitable orbitally stable solution of the form (43) is available which can then be thought of as an electron.
11. Proton-like solitons
Now let us look for solutions of the form
for real-valued functions and . Then and need to satisfy
gives a solution for Eq. (49) with . We are now looking for a positively charged soliton, so we have purposely introduced the minus sign in the solution form (48) so as to obtain clockwise rotation in the complex plane but still allow to be taken as positive. The new solution has one positive unit of elementary charge, but its energy is larger by a factor of
Solitons in the electron field
12. An unexpected result of the model
Although Model One was set up from purely classical considerations to describe the dynamical interaction of electron and proton-like solitons, a startling and perhaps unexpected result occurs. When small oscillations in the electron field occur in the vicinity of a proton-like soliton, it turns out that, because of the nature of the coupling to the electromagnetic field, only the difference frequencies are radiated, and it will be seen that the radiation is just the familiar arc spectrum of hydrogen.
Because of the scaling (50) needed for Eq. (39), the spatial size of a proton-like soliton is much smaller (by a factor of 1836) than that of an electron-like soliton even though the magnitude of the charge is the same. Consequently, in the vicinity of a proton-like soliton, Eq. (36) with
Now let us suppose that small oscillations are excited in the
The main point of interest here is that solutions of the soliton equations lead in a natural way to Eq. (52), which of course results in frequencies of the hydrogen spectrum. For the
where the spherical harmonics are written here in terms of the associated Legendre polynomials and the usual spherical coordinates
When small oscillations in the electron field
in Eqs. (41) and (42) affect the electromagnetic field
Next, let us consider the case where small oscillations occur in two different modes in the
It should be noted that charge is not necessarily quantized in Model One, and that the small oscillations in the
Thus, we find in Model One that radiation is emitted into the electromagnetic field when small oscillations in the
13. Interpretation and summary
Although Model One, Eqs. (34) and (40)–(42), was set up to let solitons mimic the dynamical behavior of electrons and protons, it turns out that small oscillations of the electron field
Here, it should be noted that three circumstances intrinsic to Model One contribute to allow this quantum-like result to occur. First, the standard, i.e., gauge invariant, coupling of the
The emission of the hydrogen spectrum in Model One has some interesting aspects but it occurs in an unexpected context. When oscillations of the
Relativistic effects have not been considered in the above treatment of Model One since it is well-known that the Klein-Gordon equation gives wrong answers for fine-structure corrections, especially those that involve the anomalous Zeeman effect. It is contemplated that the NKG equations might be replaced with nonlinear Dirac equations in a more advanced Model Two. Such a model will be comparatively difficult to examine, however, so it seems that further study of Model One will still be of interest.