The Hamiltonian Form of the KdV Equation: Multiperiodic Solutions and Applications to Quantum Mechanics

Abstract

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

1. Introduction

Roger Penrose:

It is when the ideas of symmetry and quantum mechanics come together that we find the special subtleties that are crucial to our modern understanding of the basic forces of Nature [1].

We investigate the quantum mechanics of classical soliton equations taken in the limit of microscopic scales. We consider the Korteweg–deVries (KdV) equation, a nonlinear partial differential wave Equation (PDE), as a first example. In this investigation:

(1)

We discuss how KdV has solutions in terms of Riemann theta functions and the associated Its–Matveev formula of algebraic geometry [2], which we analytically modify to give quasiperiodic Fourier series solutions of KdV. We thus argue that soliton equations in general have solutions given by a linear superposition law, which allows us to study both wave motion and soliton (particle-like) motion using the same formulation. Indeed, a simple modular transformation allows one to study nonlinear motion either in terms of oscillatory waves or in terms of solitons using the quasiperiodic Fourier modes. Clearly both sets of basis modes (waves and particles) provide the same solution to KdV. The particle and wave formulations thus occur for both the classical and quantum problems. The importance of quasiperiodicity for constructing nonlinear wave motions cannot be overemphasized; the same can also be said for numerical modeling. This contrasts with the periodic Fourier transform, which cannot keep track of coherent structures, such as Stokes waves, solitons, breathers, superbreathers, vortices, kinks, etc., while the quasiperiodic Fourier series can. This remarkable feature of quasiperiodic Fourier series means that we simultaneously have a linear superposition law and solitons in the same formulation.

(2)

We write the Hamiltonian form of KdV and show that the generalized coordinates and momenta also have quasiperiodic Fourier series solutions, an assumption first made by Heisenberg [3], Born [4] and Jorden [5,6,7] to derive matrix mechanics (see the Appendix in Heisenberg [3] and cited references).

(3)

The Hamiltonian formulation is shown to lead to matrix mechanics for KdV, including the soliton dynamics and their nonlinear interactions.

(4)

We discuss the fact that the vacuum or ground state of both the classical and quantum mechanical problems exhibit temporal time behavior and can never be set to zero. These are referred to as infra gravity waves in the classical problem.

(5)

The study of the KdV equation as given herein could be called the quantum Fermi–Pasta–Ulam (qFPU) problem, whose original study [8] led to the discovery of the soliton [9] and the inverse scattering transform (IST) for solving soliton equations [10]. In the present paper we study the qFPU problem from the point of view of temporally quasiperiodic boundary conditions for the IST, which employs our understanding of quantum finite gap theory [11,12,13,14].

(6)

Numerical modeling of KdV dynamics, both classical and quantum, is discussed as an exercise of periodic/quasiperiodic inverse scattering theory. While the classical problem is shown to have made substantial progress, the quantum problem still faces significant but straightforward challenges.

(7)

Application of the present methods is also discussed in terms of future algorithms for quantum computers. In particular, the quantum quasiperiodic Fourier transform (QqFFT) is of particular interest in the computer modeling of the quantum problem, here an exercise in matrix mechanics, whose programing we defer to a future paper.

(8)

Finally, it is remarkable that quasiperiodic Fourier series, even though they provide a linear superposition law for the study of nonlinear wave physics, also allow for the construction of the nonlinear modes in soliton theory. Nonlinearity is a natural manifestation of quasiperiodic Fourier series, while periodic Fourier series cannot codify the soliton modes. But it should be kept in mind that it is the quasiperiodicity property of finite gap theory that brings nonlinearity down from classical scales to quantum scales. One might therefore think that quantum scales would have memory of this classical nonlinearity in their formulation, even though quantum mechanics is formally linear. We show how this nonlinearity is retained at all scales, from classical to quantum, characterized in terms of the Riemann spectrum.

(9)

Nonlinearity due to perturbations of the classical KdV equation that are sufficiently high that the toroidal dynamics are destroyed can lead to chaotic dynamics at classical and quantum scales. This issue is beyond the scope of the present paper, the results of which essentially lie on tori, where integrability lives.

Here we briefly note the importance of symmetry in the present work. Both the classical results used here for the Korteweg–deVries equation and for the quantum mechanical results have been studied for their symmetry properties by a number of authors over the past 100 years. In particular, the book by Miwa, Jimbo and Date [15] gives in algebraic terms the symmetries of the KdV equation, and these lead to the particular commutation relations of quantum mechanics in terms of fermions and bosons. Of direct interest to the present paper, the occurrence of the algebro-geometric solutions of the KdV equation [12,13,14] gives the important result that the Riemann matrix is itself symmetric. Furthermore, by introducing the Hamiltonian form of the KdV equation [11,16], we can find the commutation relations that naturally occur in the development of matrix mechanics fully in terms of the Riemann matrix. The symmetries in this paper are virtually limitless: This is because the classical KdV equation has so many symmetries that it required a book to do them justice, that of Miwa, Jimbo and Date [15]. Furthermore, the quantum mechanics problem also has a huge number of symmetries, as discussed in the book by Zee [1]. For example, the matrix mechanics of Heisenberg, Born and Jorden employs SU(2), a symmetry of great importance. We note in several places throughout the important role that symmetries play in this problem.

2. The KdV Equation and Its Linear Superposition Law

The KdV equation [17] describes unidirectional shallow-water waves, which in dimensional form is given by

$$u_t+c_0{u}_x+\alpha u{u}_x+\beta u_{xxx}=0$$

(1)

KdV is also at the foundation of the Fermi–Pasta–Ulam problem [8], which led to the discovery of the inverse scattering transform [10]. We consider the solution of KdV in terms of the Cauchy problem, for which we assume that the spatial initial conditions at time $t=0$, $u\left(x,0\right)$, is given, and the evolution is then found for all space and time, $u\left(x,0\right)⟹u\left(x,t\right)$. The coefficients of (1) have the usual expressions for shallow-water waves:

$$c_0=\sqrt{gh},\alpha ={\displaystyle \frac{3c_0}{2h}},\beta ={\displaystyle \frac{c_0{h}^2}{6}}$$

(2)

where $c_0$ is the linear phase speed, h is the water depth and $g$ is the acceleration of gravity (the mks system of units is assumed in (1) and (2)). The form of the KdV equation used by Gardner, Green, Kruskal and Miura [10] ($u_t+6u{u}_x+u_{xxx}=0$) has coefficients:

$$c_0=0,\alpha =6,\beta =1$$

(3)

And the form of the equation used by Gardner [16] ($u_t+u{u}_x+u_{xxx}=0$) has coefficients:

$$c_0=0,\alpha =1,\beta =1$$

(4)

Finally, the coefficients used by Zabusky and Kruskal [9] ($u_t+u{u}_z+{\delta }^2{u}_{xxx}=0$) to discover the soliton were

$$c_0=0,\alpha =6,\beta ={\delta }^2$$

(5)

We will use (1) throughout, and thus it is easy to obtain and compare results with the other forms of KdV using (2)–(5).

The quasiperiodic solution of KdV has the form

$$u\left(x,t\right)={\displaystyle \frac{2}{\lambda }}{\partial }_{xx}\ln \theta \left(x,t\right)$$

(6)

where the second derivative of the logarithm of the Riemann theta function (6) is called the Its–Matveev formula [2]. Here the solution of KdV is scaled by the parameter $\lambda =\alpha /6\beta$ which keeps the solution in dimensional, physical units of meters and seconds.

For decades the form of the Its–Matveev formula in (6) precluded applications of the finite gap method [2,12,13,14] for the study of stochastic solutions of the KdV equation [18]. It is for this reason that Osborne [19] sought a solution to this problem by writing the right-hand side of (7) as a quasiperiodic Fourier series which is physically equivalent to the Its–Matveev formula on the left:

$$\boxed{u(x,t)={\displaystyle \frac{2}{\lambda }}{\partial }_{xx}\ln \theta \left(x,t\right)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{u}_{\mathbf{n}}{e}^{i\mathbf{n}·\mathbf{k}x-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}}$$

(7)

The quasiperiodic Fourier series on the right of (7) is a generic linear superposition law for soliton equations. The Riemann theta function has the form

$$\theta \left(x,t\right)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{\theta }_{\mathbf{n}}{e}^{i\mathbf{n}·\mathbf{k}x-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }},{ \theta }_{\mathbf{n}}=e^{-{\displaystyle \frac{1}{2}}\mathbf{n}·\tilde{\mathbf{B}}\mathbf{n}}$$

(8)

where $\tilde{\mathbf{B}}$ is the Riemann matrix. Equation (7) is a remarkable result for at least two reasons. First, the Its–Matveev formula is a tour de force of algebraic geometry [14], a miracle of modern mathematical methods, relating solutions of KdV to the Riemann theta function. Second, the corresponding quasiperiodic Fourier series is fundamentally important because it says that, in spite of the nonlinear behavior, the solutions of KdV can always be written by a linear superposition law. Thus, we have the general soliton solutions of the KdV equation which may be written with a quasiperiodic Fourier series instead of the more traditional infinite-line nonlinear soliton interactions. We refer to the linear superposition law as a spectral solution of the KdV equation which is defined by the Riemann matrix: The diagonal elements are Stokes waves and the off-diagonal elements are pair-wise interactions between the Stokes wave pairs. Stochastic solutions of the KdV equation assume that the phases in the theta function or the quasiperiodic Fourier series (7) are uniformly distributed random numbers. Each set of random numbers gives a different realization of the solutions of KdV. This result means that both data analysis and nonlinear wave modeling can be studied. Correlation functions, power spectra and coherence functions can be easily computed, just as for a linear wave equation.

In the soliton limit the Stokes waves become solitons and the off-diagonal elements give the nonlinear phase-shift interactions. The result (7) evidently generally holds true for all integrable soliton equations. Furthermore, the Fourier coefficients, $u_{\mathbf{n}}$, are given explicitly in terms of the Riemann spectrum (the Riemann matrix written in terms of ${\theta }_{\mathit{n}}=e^{-{\displaystyle \frac{1}{2}}\mathbf{n}·\tilde{\mathbf{B}}\mathbf{n}}$) [19]:

$$u_{\mathbf{n}}=-2(\mathbf{n}·\mathbf{k})\sum _{\mathbf{m}\in {\mathbb{Z}}^N}(\mathbf{n}·\mathbf{k}){\theta }_{\mathbf{m}}{\theta }_{\mathbf{n}-\mathbf{m}}^{-1}=-2(\mathbf{n}·\mathbf{k})\sum _{\mathbf{m}+\mathbf{l}=\mathbf{n}\in {\mathbb{Z}}^{\mathrm{N}}}(\mathbf{n}·\mathbf{k}){\theta }_{\mathbf{m}}{\theta }_{\mathbf{l}}^{-1}$$

(9)

Here the first equation corresponds to an inverse convolution constructed from the Its–Matveev formula and the second corresponds to three-wave interactions, well known to describe KdV behavior. The following expressions for the coefficients of the theta function hold, which with (9) and (7) connect the Riemann spectrum $\tilde{\mathbf{B}}$ to the measurable field, $u(x,t)$:

$${\theta }_{\mathbf{n}-\mathbf{m}}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\left(\mathbf{n}-\mathbf{m}\right)·\tilde{\mathbf{B}}\left(\mathbf{n}-\mathbf{m}\right)/2\right),Q_{\mathbf{m}}={\theta }_{\mathbf{m}}^{-1}=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{\left\{{\mathrm{e}}^{-(\mathbf{m}-\mathbf{n})·\tilde{\mathbf{B}}(\mathbf{m}-\mathbf{n})}\right\}}^{-1}{\delta }_{\mathbf{n}}$$

(10)

For

$$\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{\theta }_{\mathbf{n}-\mathbf{m}}{Q}_{\mathbf{m}}={\mathsf{\delta }}_{\mathbf{n}},\mathsf{\delta }=\left\{{\delta }_{\mathbf{n}}\right\}=\left[\begin{array}{c}⋮\\ 0\\ 1\\ 0\\ ⋮\end{array}\right]$$

(11)

Notice that in the first part of (11) we are considering the product of two quasiperiodic Fourier series (resulting from an infinite dimensional matrix times an an infinite length vector), such that their product is 1, here written on the right of (11) as an infinite vector whose only nonzero component is 1, found at the central postion.

This closes the connection between the general spectral solutions of KdV using finite gap theory [2] and the linear superposition law on the right of (7), which solves the KdV equation for the Cauchy problem, just as does the Its–Matveev formula. Of course the solutions (7) can be reduced to spatially periodic boundary conditions, as discussed below. Such solutions can also be viewed as “strings.” This gives an interpretation of (7) as an “infinite string” because it is quasiperiodic, not periodic.

The nonlinear dispersion relation for the KdV equation can be obtained by inserting the quasiperiodic Fourier series (7) into the KdV Equation (1). One finds

$${\hat{\mathit{\omega }}}_{\mathbf{n}}=\mathbf{n}·\mathsf{\omega }=c_0\mathbf{n}·\mathbf{k}-\beta {(\mathbf{n}·\mathbf{k})}^3+\alpha \sum _{\mathbf{m}\in {\mathbb{Z}}^N}(\mathbf{m}·\mathbf{k})u_{\mathbf{m}}{u}_{\mathbf{n}-\mathbf{m}}/u_{\mathbf{n}}$$

(12)

Here the nonlinear convolution can also be written as three-wave interactions:

$$\sum _{\mathbf{m}\in {\mathbb{Z}}^N}(\mathbf{m}·\mathbf{k})u_{\mathbf{m}}{u}_{\mathbf{n}-\mathbf{m}}=\sum _{\mathbf{m}+\mathbf{l}=\mathbf{n}\in {\mathbb{Z}}^N}(\mathbf{m}·\mathbf{k})u_{\mathbf{m}}{u}_{\mathbf{l}}$$

(13)

The point set of frequencies, $\mathbf{n}·\mathsf{\omega }$ (12), is often referred to as a discretuum [20] because, while discrete, the point set is infinitely dense, essentially simultaneously approximating a continuum. The discretuum frequencies are Diaphantine and are therefore unique. We discuss the discretuum in more detail below. The algebraic geometric form of the dispersion relation, equivalent to (12) and (13), is discussed in [2], where the algebraic geometric loop integrals are given. A modern summary is given by [14].

3. The Spatially Periodic Solutions of the KdV Equation

It is often useful to assume spatially periodic boundary conditions for the KdV equation. For example, Zabusky and Kruskal [9] used these to develop the concept of the soliton in nonlinear wave theory. Imposing spatially periodic boundary conditions on $(0<x<L)$ means that

$$u\left(x,t\right)=\sum _{n=-\infty }^{\infty }{u}_n\left(t\right)e^{i{k}_{n}x}, k_n=2\pi n/L$$

(14)

where we have introduced a time-dependent Fourier coefficient, $u_n\left(t\right)$. Inserting (14) into the KdV Equation (1) (a PDE) gives a set of ordinary differential Equation (15) (ODEs) for the time evolution of the coefficients in the Fourier series:

$${\stackrel{.}{u}}_n+i{\omega }_{0n}{u}_n+i\alpha \sum _{m=-\infty }^{\infty }{k}_m{u}_m{u}_{n-m}=0$$

(15)

The over dot means time derivative. Here the linear dispersion relation is given by (set $\alpha =0$)

$${\omega }_{0n}=c_0{k}_n-\beta k_n^3$$

(16)

The ODEs, Equation (15), can be written in a more convenient form:

$${\stackrel{.}{u}}_n+i{\hat{\omega }}_n{u}_n=0$$

(17)

where the associated nonlinear dispersion relation for the spatially periodic KdV equation thus has the following form, as we can see from (12):

$${\hat{\mathsf{\omega }}}_n={\hat{\mathsf{\omega }}}_{k_n=\mathbf{n}·\mathbf{k}}=\mathbf{n}·\mathsf{\omega }=c_0\mathbf{n}·\mathbf{k}-\beta {(\mathbf{n}·\mathbf{k})}^3+\alpha \sum _{\mathbf{m}\in {\mathbb{Z}}^N}(\mathbf{m}·\mathbf{k})u_{\mathbf{m}}{u}_{\mathbf{n}-\mathbf{m}}/u_n$$

(18)

It is clear that the Its–Matveev formula and its associated quasiperiodic Fourier series (7) can be reduced to the periodic Fourier series (14), and the associated ODEs for the Fourier coefficients (15) for spatially periodic boundary conditions, such that $k_n=2\pi n/L$, and simultaneously the time-varying Fourier coefficient ODEs (15) have a quasiperiodic Fourier series solution, which we write as follows [19]:

$$u_n(t)=\sum _{\{\mathbf{m}\in {\mathbb{Z}}^N: k_n=\mathbf{n}·\mathbf{k}\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}$$

(19)

It should be clear that the quasiperiodic Fourier series (19) solves the ODEs (15), which can be verified by a simple substitution. In summary: (19) substituted into (14) with the dispersion relation (18) solves the KdV Equation (1) for spatially periodic boundary conditions. The expression (12), as written in the more explicit form $\mathbf{n}·\mathbf{k}⟹k_n$ (18), occurs because we have temporally quasiperiodic boundary conditions. In this case we can see that $\mathbf{n}·\mathbf{k}⟹k_n$ because the wave numbers are commensurable (corresponding to spatial periodicity). Each dot product reduces to a wave number in the “basis set” of commensurable wave numbers, $\mathbf{k}=[k_1,k_2...k_N]$, $k_n=\mathbf{n}·\mathbf{k}=2\pi n/L.$ The $\mathbf{n}·\mathsf{\omega }$ instead lie on the discretuum because they are incommensurable. The points of the discretuum are known to be Diaphantine and are therefore unique, never overlapping.

It should be clear that the above considerations provide a simple recipe for numerical computations for summing the Riemann theta functions and the quasiperiodic Fourier series, results which are essential for numerical modeling of the KdV equation. Traditional numerical modeling treats the “model” as Equations (14) and (15). A fast Fourier transform computes (14), while a Runge Kutta algorithm is traditionally used to compute (15). A numerical model using the methods given herein is of a different type. We can see this by noting that the Its–Matveev formula and the quasiperiodic Fourier series solution to the KdV Equation (7) can be thought of as “numerical models” for solving the classical KdV equation. See the details for computing the Its–Matveev formula in Chapter 32 in [15]. Likewise, the quasiperiodic Fourier series solution to KdV can also be used as a numerical model. Furthermore, the quasiperiodic Fourier series can be used to Fourier analyze shallow-water ocean wave data for its nonlinear spectral content, i.e., for finding the Stokes waves and solitons in the spectrum of a measured time series. See Figure 1, Figure 2 and Figure 3, below, for examples of nonlinear spectra.

To better understand the notation given here and the relation to numerical modeling, let us graph some of the spectral properties for the wavenumbers and frequencies. In the linear case, where $\alpha =0$, we have the spectra for the wavenumbers, which are assumed to be commensurable, and for the frequencies, which, by (18), are incommensurable and given approximately by $\omega \simeq c_{o}k-\beta k^3+\gamma a^2$ (a is the Fourier mode amplitude and $\gamma =9c_o/16k{h}^4$), as shown in the lower panel of Figure 1. Note that the $\gamma a^2$ term comes from the nonlinear convolution term in (18) for a single degree of freedom.

Because the wavenumbers are commensurable, their dot products with the integer summation vecters, n, lie on the same set of commensurable wavenumbers: $\mathbf{n}·\mathbf{k}⟹k_n$, although each of these basis wavenumbers, $k_n=\mathbf{n}·\mathbf{k}=2\pi n/L$, is repeated an infinite number of times. As a consequence, the frequencies are incommensurable and unique and given by (18): As stated above the frequencies $\mathbf{n}·\mathsf{\omega }$ fall on a discretuum of points. Thus, while they are discrete and unique, they are nevertheless so densely packed as to blacken the spectral domain. We show graphically some of these frequencies, (partially) filling the region between the base frequencies (red), in Figure 2.

Another convenient way to represent the nonlinear spectrum of a quasiperiodic Fourier series is shown in Figure 3. These are the “coherent modes” of the spectrum which consist of the Stokes waves and in the large amplitude limit are solitons. The Stokes waves contain the higher-order harmonics, graphed perpendicular to the usual frequency axis and labeled the “bound frequencies”. The frequency axis itself refers to the “free” Stokes modes. Where is the discretuum shown in Figure 2 to be found in Figure 3? It lies on the frequency axis and densely fills the spaces between the Stokes waves, thus constituting the “Stokes wave interactions.” In the soliton limit of the Stokes waves, these interactions construct the “soliton phase shifts” as normally derived from infinite-line soliton theory. The soliton behavior of (14) and (19) provides a modern approach to duplicate the numerical experiments of Zabusky and Kruskal [9], in which the soliton was discovered. As a numerical model, (14) and (19) have been referred to as “hyperfast” because of the enhanced speed of the algorithm with respect to traditional methods (see Chapter 32 of Osborne [18]).

Here is an example of (7) for N degrees of freedom written as a nested summation:

$$\begin{array}{c}u\left(x,t\right)={\displaystyle \sum _{\mathbf{n}\in {\mathbb{Z}}^N}}{u}_n{\mathrm{e}}^{\mathrm{i}\mathbf{n}·\mathbf{k}x-\mathrm{i}\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}\\ ={\displaystyle \sum _{n_1=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \sum _{n_2=-\mathrm{\infty }}^{\mathrm{\infty }}}\dots {\displaystyle \sum _{n_N=-\mathrm{\infty }}^{\mathrm{\infty }}}{\mathrm{u}}_{n_1,n_2...n_N}{\mathrm{e}}^{i{\sum }_{\mathrm{m}=1}^N{n}_{\mathrm{m}}{k}_{\mathrm{m}}x-i{\sum }_{m=1}^N{n}_m{\omega }_{m}t-i{\sum }_{m=1}^N{n}_m{\varphi }_m}\end{array}$$

(20)

This expression is just a representation of a genus, N, solution of the KdV Equation (there are N nested sums), i.e., there are N modes consisting generally of a mixture of sine waves, Stokes waves and solitons, depending on how nonlinear each mode is. Each mode corresponds to an individual summation in (20). The case for two degrees of freedom is given by

$$u\left(x,t\right)=\sum _{n_1=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{n_2=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_{n_1,n_2}{\mathrm{e}}^{i{\sum }_{m=1}^2 n_m{k}_{m}x-i{\sum }_{m=1}^2{n}_m{\omega }_{m}t-\mathrm{i}{\sum }_{m=1}^2{n}_m{\varphi }_m}$$

(21)

This expression may be written as the sum of two solitons plus the nonlinear (phase-shift) interactions. Of course by a soliton we mean the Stokes series for a single summation, provided that the elliptic modulus for each summation is sufficiently near 1.

A single summation from the quasiperiodic Fourier series is given by

$$u\left(x,t\right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_n{\mathrm{e}}^{\mathrm{i}nkx-in\omega \left(k\right)t+in\varphi }$$

(22)

This expression corresponds to a single Stokes wave: When the amplitude is small, we have a sine wave. Intermediate size amplitudes give a traditional Stokes wave, and if the amplitudes are large enough we obtain a soliton. Equations (9)–(11) determine how nonlinear each Stokes mode is and how strong their pair-wise interactions are. A single Riemann matrix diagonal element represents a soliton when the matrix element is sufficiently small and its elliptic parameter is thus sufficiently near 1 [21]. The connection of the Riemann matrix elements and the soliton properties are given by (9)–(11) for the computations of the coefficients $u_{n_1,n_2\dots n_N}$ and for the amplitude-dependent dispersion relation (18).

We now provide a number of simple calculations to aid in physical understanding and numerical modeling efforts. The so-called elliptic function solution of the KdV equation, or cnoidal wave, corresponds to a single Riemann matrix diagonal mode and is given in dimensional units by [21]:

$$\eta (x,t)=2{\eta }_{0}c{n}^2\left\{\right(K\left(m\right)/\pi \left)\right[k(x-Ct+\varphi )]\mid m\}$$

Here $K\left(m\right)$ is the elliptic integral, which is related to the Ursell number, $U$, by

$$m{K}^2\left(m\right)={\displaystyle \frac{3{\pi }^2{\eta }_0}{2k^2{h}^3}}=4{\pi }^{2}U, U={\displaystyle \frac{3{\eta }_0}{8k^2{h}^3}}$$

This gives the higher-order expression for the Ursell number in terms of the nome, q:

$$q(1+q^2+2q^4+5q^6+14q^8+\dots )={\displaystyle \frac{3a}{4h^3{k}^2}}=U$$

The Ursell number is the most important physical, dimensionless parameter in classical KdV dynamics. In the above formulas k is the wavenumber, ${\eta }_0$ is the cnoidal wave amplitude, C is the phase speed, m is the elliptic modulus and q(m) is the elliptic nome, where $q=exp(-B/2)$ and B is the element of the 1 × 1 Riemann matrix for a single-degree-of-freedom solution of the KdV equation. Because $q\approx U$, for a small amplitude Stokes wave (see the last equation, above), the Ursell number is related to the diagonal elements of the Riemann matrix by $U\approx exp(-B/2)$. If one chooses a set of Ursell numbers in this case for each diagonal element of the Riemann spectrum, then only the off-diagonal elements remain to be computed to complete the spectral matrix for a numerical calculation. These may be determined by Poincare series as described in Osborne [18], Chapter 32.

The phase speed has the form

$$C=\omega /k=c_0\{1+2\eta /h-2k^2{h}^2{K}^2(m)/3{\pi }^2\}$$

The Fourier series for a single cnoidal wave is given by

$$u(x,t)=\lambda \eta (x,t)=2{\partial }_{xx}\mathrm{l}\mathrm{n}{ \theta }_3(x,q)=4k^2\sum _{n=1}^{\infty }{\displaystyle \frac{{\left(-1\right)}^{n}n{q}^n}{1-q^{2n}}}\mathrm{c}\mathrm{o}\mathrm{s}[n(kx-\omega t+\varphi )]$$

Here ${\theta }_3(x,t|q)$ is the third Jacobi elliptic function, equivalent to the single-degree-of-freedom mode of the Riemann theta function (with a 1 × 1 Riemann matrix) as defined herein. The dispersion relation is given by

$$\begin{array}{c}\omega =c_{0}k\left\{1+{\displaystyle \frac{2\eta }{h}}-{\displaystyle \frac{2k^2{h}^2{K}^2\left(m\right)}{3{\pi }^2}}\right\}\\ \\ \\ \omega \approx c_{0}k-\beta k^3[1-24q^2-72q^4-96q^6-168q^8+...]\end{array}$$

These results define the the Fourier coefficients and parameters in (22):

$$u_n=4k^2{\displaystyle \frac{(-1{)}^{n}n{q}^n}{1-q^{2n}}}$$

The nome, $q=e^{-{\displaystyle \frac{1}{2}}B}$, is a more practical parameter than the elliptic modulus, m(q), for numerical computations. Note that the above coefficient equation can be used to solve for the nome, q, in terms of the Fourier Stokes mode, $u_n$: Each of these gives a diagonal element of the Riemann matrix. The off-diagonal elements can be computed by Poincare series [18]. Using these simple ideas means that one can take the quasiperiodic Fourier transform of time-series data and then the Riemann matrix can easily be found, thus specifying the character of the nonlinear modes. This can happen because integrability allows one to compute all aspects of the solutions of the soliton equations. Please note that we deal exclusively with integrable soliton equations in this and later studies.

A single “Stokes wave” to third order, in physical parameters, is given by the expansion of the above Fourier series [21]:

$$u\left(x,t\right)=a \mathrm{c}\mathrm{o}\mathrm{s}X+{\displaystyle \frac{3a^2}{4k^2{h}^3}}\mathrm{c}\mathrm{o}\mathrm{s}2X+{\displaystyle \frac{27a^3}{64k^4{h}^6}}\mathrm{c}\mathrm{o}\mathrm{s}3X+...$$

$$X=kx-\omega t+\varphi , \omega =c_{0}k-\beta k^3+\gamma a^2, \gamma ={\displaystyle \frac{9c_{\mathrm{o}}}{16k{h}^4}}$$

Note that the dispersion relation $\omega =c_{0}k-\beta k^3+\gamma a^2$ has a nonlinear “Stokes frequency correction” proportional to the squared amplitude, $\gamma a^2$, which is easily computed from (18) for a single Stokes wave. These results can be used to check the numerical computations and interpret the quasiperiodic Fourier Stokes modes in Figure 3. The Stokes mode amplitudes have been graphed in Figure 3 as a function of the “bound modes,” i.e., the individual phase-locked Fourier modes of the Stokes wave. When the elliptic parameter, m, asymptotes to 1 and q naturally and simultaneously tends to 1, the Stokes wave becomes a soliton. When two adjacent Stokes waves are phase-locked with one another, the associated coherent structure is a “nonlinear beat” or “breather.” “Superbreathers” occur for phase locking of more than two adjacent Stokes modes.

The word “breather” is perhaps more appropriate for the focusing nonlinear Schrödinger equation which occurs in deep water, where the nonlinear beat has space–time amplitude dynamics generated by the modulational instability, i.e., the packet breathes up and down during its evolution. The phase locking is dynamic and automatic in deep water, but for the shallow-water KdV equation the phase locking usually occurs first in deep water and then maintains phase locking as the wave trains propagate into shallow water, where the breathing cycle no longer occurs but the large packet behavior continues, which we then refer to as a “fossil breather,” i. e., a large nonlinear wave packet continues to propagate in shallow water with “memory” of its past creation. This memory should be discoverable by the numerical quasiperiodic Fourier series analysis of time-series data, i.e., phase locking between two Stokes waves. Fossil breathers in shallow water lead to breakers that surfers are famous for enjoying.

An important limit is that for which all the modes in the Riemann matrix tend to solitons. In this case the theta function has an alternative form which converges much more rapidly, a so-called Gaussian series:

$$\theta (x,t)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}\mathrm{e}\mathrm{x}\mathrm{p}[-{\displaystyle \frac{1}{2}}(\mathbf{k}x-\mathsf{\omega }t+\mathsf{\varphi }-\pi \mathbf{n})·\mathbf{F}(\mathbf{k}x-\mathsf{\omega }t+\mathsf{\varphi }-\pi \mathbf{n})]$$

This latter expression is useful when the only modes in the system are solitons, a case we will study for the quantum problem in the near future.

It is further worthwhile looking at the details for a particular quasiperiodic Fourier series, a two-degree-of-freedom system, which we write as follows (see Equation (21)):

$${\theta }_{22}(X_1,X_2)=\sum _{n_1=-2}^2\sum _{n_2=-2}^2{q}^{n_1^2}{p}^{n_2^2}{r}^{n_1{n}_2}\mathrm{c}\mathrm{o}\mathrm{s}(n_1{X}_1+n_2{X}_2)$$

For simplicity we use here the quasiperiodic Fourier series for the Riemann theta function. For additional simplicity we have also assumed limits for the two nested sums to be −2, 2. We graph in

Table 1. Theta function partial sum for M = 2, N = 2.

, below, all the terms in the resultant “partial theta summation.” The Riemann matrix element corresponds to two degrees of freedom, with the Riemann matrix being 2 × 2 (see the general form (8)). There are 25 terms in the partial theta summation and each is identified in the table. The pairs of summation integers $(n_1,n_2)$ are given in the third column. An ordering parameter, j, runs from −12, −11… −1, 0, 1… 11, 12 and is given in the second column. Each of the summation terms $q^{n_1^2}{p}^{n_2^2}{r}^{n_1{n}_2}\mathrm{c}\mathrm{o}\mathrm{s}(n_1{X}_1+n_2{X}_2)$ is given in the fourth column. By abuse of notation, the term $q=exp(-B_{11}/2)$ (the first Stokes wave) corresponds to the first diagonal element of the Riemann matrix, while $p=\mathrm{e}\mathrm{x}\mathrm{p}(-B_{22}/2)$ (the second Stokes wave) corresponds to the second diagonal element and $r=\mathrm{e}\mathrm{x}\mathrm{p}(-B_{12})$ corresponds to the off-diagonal element which governs the interactions between the two Stokes waves. The qs and ps used in this section are defined in terms of the Riemann matrix and are not the generalized coordinates discussed below for the Hamiltonian formulation.

In Figure 4 we give a graph of the pairs of summation integers, $\mathbf{n}=(n_1,n_2)$, as a function of the ordering parameter, j (on the horizontal axis at the bottom). On the left vertical axis are the commensurable wavenumbers $I_l=\mathbf{n}·\mathbf{k}⟹k_n$, which run from −7, −6… to 1, 0, 1… 6, 7. The zig-zag line shows the actual terms summed, as given in Table 1. Each intersection of the zig-zag line corresponds to the crossing of a commensurable wavenumber $I_l=\mathbf{n}·\mathbf{k}⟹k_n$. Notice that for $I_l=\mathbf{n}·\mathbf{k}⟹k_n=0$, we have the “ground state” (corresponding to the three red rectangles of Table 1) of the two-by-two quasiperiodic Fourier series: Note that there are three terms corresponding to Table 1, the three red dots, that contribute to the series for the ground state. Also note that for $I_l=\mathbf{n}·\mathbf{k}⟹k_n=2$ (in this Section we assume $L=2\pi$) there are three blue dots that contribute three terms to the series. For $I_l=\mathbf{n}·\mathbf{k}⟹k_n=-2$, there are another three blue dots that contribute an additional three sine waves to the quasiperiodic Fourier series.

Of course, any realistic summation for an actual application will contain hundreds of thousands of terms, but such a case is not easily graphable, so we used fewer terms in Figure 4. Now let us consider some examples of terms in this simple partial summation.

• Simple Example #1

We can see from Figure 4 that for the zero wavenumber, $k=0$, there are three integer vectors, (−2,1), (0,0) and (2,−1), which give three terms that contribute to the Fourier series $u_0\left(t\right)$, the zeroth term in (14), which is the ground or vacuum state. The three frequencies are computed from

$${\Omega }_{\mathrm{n}}=\mathbf{n}·\mathsf{\omega }=n_1{\omega }_1+n_2{\omega }_2\ll 1$$

For which

$${\Omega }_{-\mathrm{2,1}}=n_1{\omega }_1+n_2{\omega }_2=-2{\omega }_1+{\omega }_2\ll 1$$

$${\Omega }_{\mathrm{0,0}}=n_1{\omega }_1+n_2{\omega }_2=0$$

$${\Omega }_{2,-1}=n_1{\omega }_1+n_2{\omega }_2=2{\omega }_1-{\omega }_2\ll 1$$

The three Fourier modes are shown in the three red boxes in Table 1 and the single red box in Figure 4. Thus, in the present simple case the single wavenumber $k=0$ corresponds to three frequencies. This result looks like line splitting in classical quantum mechanics, which is here true for the KdV nonlinear problem, but the results appear linear (they have indeed been linearized) for the present example:

Thus, the frequencies look like a line spectrum, the kind that scientists were studying in the 1920s (or, for that matter, which Newton studied in his investigation of optics in a prism). This case for $k_0=0$ corresponds to the zero or ground state, which is responsible for infra gravity waves in shallow-water ocean waves.

• Simple Example #2

We can see from Figure 4 that for wavenumbers $k=\pm 2$ (see line splitting column) there are six integer vectors, (−2, 0), (0, −1), (2, −2), (−2, 2), (0, 1) and (2, 0) (all colored blue), which give six Fourier terms for the Fourier series (14) with coefficients, $u_2\left(t\right)$ and $u_{-2}\left(t\right)$, all of which make finite contributions.

Then the six frequencies are computed and found to be, first for $k=2$:

$${\Omega }_{-\mathrm{2,2}}=n_1{\omega }_1+n_2{\omega }_2=-2({\omega }_1-{\omega }_2)$$

$${\Omega }_{\mathrm{0,1}}=n_1{\omega }_1+n_2{\omega }_2={\omega }_2$$

$${\Omega }_{\mathrm{2,0}}=n_1{\omega }_1+n_2{\omega }_2=2{\omega }_1$$

And for $k=-2$:

$${\Omega }_{-\mathrm{2,0}}=n_1{\omega }_1+n_2{\omega }_2=-2{\omega }_1$$

$${\Omega }_{0,-1}=n_1{\omega }_1+n_2{\omega }_2=-{\omega }_2$$

$${\Omega }_{2,-2}=n_1{\omega }_1+n_2{\omega }_2=2({\omega }_1-{\omega }_2)$$

The six Fourier modes are shown in the six blue boxes in Table 1 and the two blue boxes in Figure 4. Thus, in the present simple case the single wavenumber $k=2$ corresponds to three frequencies:

And for $k=-2$ we have three more:

From the last example we can see that we observe line splitting for an arbitrary selection of wave numbers in (14). From these simple examples we can see that each wavenumber Fourier component, $u_n\left(t\right)$, corresponds to each commensurable wavenumber, $k_n=\mathbf{n}·\mathbf{k}$ (formally, there is an infinite number of each of them, not just three as in the example of Table 1, Figure 4), and hence they there are arbitrarily large numbers of frequency Fourier modes in any realistic physical situation (essentially contributing to the discretuum), which we write as $\mathbf{n}·\mathsf{\omega }$. Typically, in a classical simulation of KdV we have several hundred thousand of these frequencies, which represent the discretuum.

4. The Hamiltonian Form of the KdV Equation Has Generalized Coordinates and Momenta Which Are Quasiperiodic Fourier Series

If the spectral solutions of the KdV equation are quasiperiodic Fourier series, as in the second equation of (7), then is it possible that the generalized coordinates, q, and momenta, p, of the Hamiltonian formulation of KdV are also quasiperiodic Fourier series? The reason that this question arises is that Born [4], Born, Heisenberg and Jordon [5]; Born and Jorden [6]; and Heisenberg [3,7] assumed “for simplicity” that the coordinates and momenta are given by quasiperiodic Fourier series (see the discussion in the Appendix of Heisenberg [3]). What followed from this assumption was the development of the matrix mechanics formulation of quantum theory. In modern work in quantum mechanics, one often uses an ordinary periodic Fourier series (as found in typical quantum mechanics text books) instead of a quasiperiodic Fourier series (as used by Heisenberg). This assumption, although it may seem reasonable if one uses linearity as the basis of quantum mechanics, eliminates the important pathway to the integrability of soliton systems through Riemann theta functions and their linear superposition law derived from the Its–Matveev formula. Quasiperiodicity, remarkably, leads to the nonlinear soliton dynamics of integrable, classical wave equations and to nonlinearity in quantum mechanics.

Thus, should the Hamiltonian generalized coordinates and momenta for the KdV equation be quasiperiodic Fourier series, then the development of matrix mechanics of the KdV equation would then follow in a natural way as described by Heisenberg. We would therefore have quantum mechanics (a linear theory) exactly valid for nonlinear classical soliton equations! Thus, our understanding of the quantization of nonlinear, classical wave equations would be improved, just by using our knowledge of finite gap integration theory for soliton equations. Let us see if this is indeed the case.

The eventual goal of this work is to develop a kind of nonlinear quantum field theory (which we refer to as the quantum FPU problem) by first capturing the nonlinear dynamics of an integrable classical field equation using finite gap theory and then to quantize the associated soliton dynamics through its Hamiltonian formulation.

In the Appendix of his book, Heisenberg [3] suggests that one should address the Hamiltonian form of a classical physics problem and thereby seek the associated quantum mechanics. He assumes “for simplicity” that the generalized coordinates and momenta have quasiperiodic Fourier series:

$$q_n(t)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{q}_{\mathbf{n}}^n{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}$$

(23)

$$p_n(t)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{p}_{\mathbf{n}}^n{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}$$

(24)

where $n=\mathrm{1,2}...N$ are the spatial Fourier modes. Our modern perspective is that for soliton systems, quasiperiodicity implies integrability of the classical wave equation. It is amazing that Heisenberg selected soliton integrability in his “simple” solution of Hamilton’s equations and from this developed matrix mechanics! He made this remarkable step fifty years before quasiperiodic soliton integrability was discovered [2,12,13]. While Born, Heisenberg and Jordon were dealing with systems of particles in their development of matrix mechanics, we now show that the generalized coordinates and momenta also appear as quasiperiodic Fourier series for soliton field equations, provided we address the Hamiltonian formulation of the KdV equations as first developed by Gardner [16]. See also the book by Faddeev and Takhtajan [11].

We now give a terse overview of the Hamiltonian form of the KdV equation. One begins with a classical variational principle which requires the definition of a functional:

$$F[u]={\int }_{-L/2}^{L/2}f(x,u,u_x)dx$$

(25)

We assume that the spatial variable, x, is on the interval $(0,L)$ or $(-L/2,L/2)$. In this expression f is viewed as a function of x, u and $u_x$, such that $u=u\left(x\right)$ and $u_x=du\left(x\right)/dx.$ We are familiar with the fact that one normally deals with the action integral, where f is the Lagrangian and $u\left(x\right)$ is mapped to the generalized coordinate, $q\left(t\right).$ This leads to the functional derivative:

$${\displaystyle \frac{\delta F}{\delta u}}={\displaystyle \frac{\partial f}{\partial u}}-{\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{\partial f}{\partial u_x}}\right)$$

(26)

Now, keep in mind that the Hamiltonian, which is known to be related to the third conservation law of the KdV equation, has the form

$$F_3=\underset{-L/2}{\overset{L/2}{\int }}\left(-{\displaystyle \frac{c_0}{2}}{u}^2-{\displaystyle \frac{\alpha }{6}}{u}^3+{\displaystyle \frac{1}{2}}\beta u_x^2\right)dx$$

(27)

We now find from (26) and (27)

$${\displaystyle \frac{\delta F_3}{\delta u}}=-c_{0}u-{\displaystyle \frac{\alpha }{2}}{u}^2-\beta u_{xx}$$

(28)

Then the KdV Equation (1) is determined from the associated Lagrangian by

$$u_t={\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{\delta F_3}{\delta u}}\right)=-c_0{u}_x-\alpha u{u}_x-\beta u_{xxx}$$

(29)

In a straightforward derivation of the time derivative of the Fourier coefficients in (14), we find the form

$${\displaystyle \frac{d{u}_n\left(t\right)}{dt}}={\displaystyle \frac{i{k}_n}{L}}{\displaystyle \frac{\partial F_3}{\partial u_{-n}}}$$

(30)

It is then convenient to define the generalized coordinates, momenta and Hamiltonian as:

$$q_n(t)={\displaystyle \frac{u_n\left(t\right)}{k_n}}$$

$$p_n\left(t\right)=u_{-n}\left(t\right)$$

(31)

$$H={\displaystyle \frac{i}{L}}{F}_3={\displaystyle \frac{i}{L}}\underset{-L/2}{\overset{L/2}{\int }}\left(-{\displaystyle \frac{c_0}{2}}{u}^2-{\displaystyle \frac{\alpha }{6}}{u}^3+{\displaystyle \frac{1}{2}}\beta u_x^2\right)dx$$

This then leads to Hamilton’s equations:

$$\begin{gathered} {\displaystyle \frac{d{q}_n}{dt}}={\displaystyle \frac{\partial H}{\partial p_n}} \\ {\displaystyle \frac{d{p}_n}{dt}}=-{\displaystyle \frac{\partial H}{\partial q_n}} \end{gathered}$$

(32)

From (19) and (31) we have the quasiperiodic Fourier series for the generalized coordinates and momenta for the KdV equation:

$$\begin{gathered} q_n(t)={\displaystyle \frac{1}{k_n}}\sum _{\{\mathbf{n}\in {\mathbb{Z}}^N: n=L\mathbf{n}·\mathbf{k}/2\pi >0\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }} \\ {p}_n(t)=\sum _{\{\mathbf{n}\in {\mathbb{Z}}^N: n=L\mathbf{n}·\mathbf{k}/2\pi <0\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }} \end{gathered}$$

(33)

Here $n=\mathrm{1,2}...N$ and $L\mathbf{n}·\mathbf{k}/2\pi =\pm n$ for n, a positive integer. We are summing over all n in this process, and when the dot product $L\mathit{n}·\mathit{k}/2\pi =+n$ we have contributions to $q_n\left(t\right)$. When the dot product $L\mathbf{n}·\mathbf{k}/2\pi =-n$ we have contributions to $p_n\left(t\right)$.

Now Equation (33) may be written as follows:

$$\begin{gathered} q_n(t)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{q}_{\mathbf{n}}^n{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }} \\ {p}_n(t)=\sum _{\mathbf{n}\in {\mathbb{Z}}^N}{p}_{\mathbf{n}}^n{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }} \end{gathered}$$

(34)

where the coefficients are related to the positive or negative integers by

$$\begin{gathered} q_{\mathbf{n}}^n={\displaystyle \frac{u_{\mathbf{n}}}{k_n}}{\delta }_{n, L\mathbf{n}·\mathbf{k}/2\pi >0} \\ {p}_{\mathbf{n}}^n=u_{\mathbf{n}}{\delta }_{n, L\mathbf{n}·\mathbf{k}/2\pi <0} \end{gathered}$$

(35)

and the Kronecker delta function is given by

$${\delta }_{n,m}=f\left(x\right)=\left\{\begin{array}{c}1, n=m\\ 0, else\end{array}\right.$$

(36)

Thus, we have found that for the KdV equation the generalized coordinates and momenta have quasiperiodic Fourier series (34), just as Heisenberg assumed.

5. Evaluation of the Gardner Formulation of the Hamiltonian for the KdV Equation

The Hamiltonian for the KdV equation, as already discussed above, is given in dimensional form by the following (see Gardner [16], Feddeev and Takhtajan [11]):

$$H={\displaystyle \frac{i}{L}}{F}_3={\displaystyle \frac{i}{L}}\underset{-L/2}{\overset{L/2}{\int }}\left(-{\displaystyle \frac{c_0}{2}}{u}^2-{\displaystyle \frac{\alpha }{6}}{u}^3+{\displaystyle \frac{1}{2}}\beta u_x^2\right)dx$$

(37)

The constant coefficients in this expression are those of the KdV Equation (1). To obtain the Hamiltonian for other forms of KdV, use the coefficients (2)–(5).

We note that many of the results of this paper are based upon the idea that one can add, subtract, multiply, divide, take derivatives and compute integrals of periodic and quasiperiodic Fourier series. The algebra and conditions for the validity of these operations are well known [22,23].

Starting from (14), we now investigate how to compute this Hamiltonian integral (37). First, let us suppose:

$$-{\displaystyle \frac{c_o}{2}}{u}^2-{\displaystyle \frac{\alpha }{6}}{u}^3+{\displaystyle \frac{1}{2}}\beta u_x^2=\sum _{n=-\infty }^{\infty }{D}_n\left(t\right)e^{i{k}_{n}x},{ k}_n={\displaystyle \frac{2\pi n}{L}}$$

(38)

where we determine $D_n\left(t\right)$ below. To compute (38) with (14) and (19), we need to momentarily let

$$H={\displaystyle \frac{i}{L}}\underset{-L/2}{\overset{L/2}{\int }}\sum _{n=-\infty }^{\infty }{D}_n\left(t\right)e^{i{k}_{n}x}dx={\displaystyle \frac{i}{L}}\sum _{n=-\infty }^{\infty }{D}_n\left(t\right)\underset{-L/2}{\overset{L/2}{\int }}{e}^{i{k}_{n}x}dx$$

(39)

Then the integral has the form

$$\underset{-L/2}{\overset{L/2}{\int }}{e}^{i{k}_{n}x}dx={\displaystyle \frac{e^{\frac{2\pi inx}{L}}}{2\pi in/L}}\left|\begin{array}{c}L/2\\ -L/2\end{array}\right.={\displaystyle \frac{L}{2}}{\displaystyle \frac{e^{\frac{2\pi inL}{2L}}-e^{-\frac{2\pi inL}{2L}}}{i\pi n}}=$$

(40)

$$={\displaystyle \frac{L}{2}}{\displaystyle \frac{e^{i\pi n}{-e}^{-i\pi n}}{\pi n}}=L{\displaystyle \frac{\sin \left(\pi n\right)}{\pi n}}=L{\delta }_n$$

(41)

Finally,

$$\underset{-L/2}{\overset{L/2}{\int }}{e}^{i{k}_{n}x}dx=L{\delta }_n$$

(42)

This is the Kronecker delta function that came from the above integral:

$${\delta }_n={\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(n\pi \right)}{n\pi }}=\left[\begin{array}{cc}1, & \mathrm{i}\mathrm{f} n=0\\ 0, & n=\mathrm{1,2}...\infty \end{array}\right]$$

(43)

for n, an integer. This is shown in the graph of Figure 5.

Use this result in Equation (39) to obtain

$$H(t)=i\sum _{n=-\infty }^{\infty }{D}_n(t){\delta }_n=i{D}_0(t)$$

(44)

This result is equivalent to taking the mean

$$\langle \left(-{\displaystyle \frac{c_o}{2}}{u}^2-{\displaystyle \frac{\alpha }{6}}{u}^3+{\displaystyle \frac{1}{2}}\beta u_x^2\right)\rangle =\sum _{n=-\infty }^{\infty }{D}_n\left(t\right)\langle e^{i{k}_{n}x}\rangle =D_0(t)$$

(45)

where the mean is given by

$$〈· 〉={\displaystyle \frac{1}{L}}\underset{-L/2}{\overset{L/2}{\int }}·dx$$

(46)

Now let us compute the individual terms in the Hamiltonian (45).

First, the squared term is a simple convolution:

$$u^2\left(x,t\right)=\sum _{m=-\infty }^{\infty }{A}_m{e}^{i{k}_{m}x}, A_m=\sum _{n=-\infty }^{\infty }{u}_n{u}_{m-n}$$

(47)

Proof. Coefficient

$A_n\left(t\right)$.

$$u^2(x,t)=\left(\sum _{l=-\infty }^{\infty }{u}_l{e}^{i{k}_{l}x}\right)\left(\sum _{m=-\infty }^{\infty }{u}_m{e}^{i{k}_{m}x}\right)=\sum _{n=-\infty }^{\infty }{A}_n{e}^{i{k}_{n}x}$$

(48)

where

$$A_n(t)=\sum _{m=-\infty }^{\infty }{u}_m{u}_{n-m}$$

(49)

is an auto (self) convolution. □

Then, the cubic term is a double convolution:

$$u^3\left(x,t\right)=\sum _{n=-\infty }^{\infty }{B}_n\left(t\right)e^{i{k}_{n}x}, B_n(t)=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}[(u_n{)}^3]$$

(50)

Proof. Coefficient$B_n\left(t\right)$.

$$u^3(x,t)=u(u^2)=\left(\sum _{l=-\infty }^{\infty }{u}_l{e}^{i{k}_{l}x}\right)\left(\sum _{m=-\infty }^{\infty }{A}_m{e}^{i{k}_{m}x}\right)=\sum _{n=-\infty }^{\infty }{B}_n\left(t\right)e^{i{k}_{n}x}$$

(51)

where from the above (49) we obtain (see Appendix A)

$$B_n=\sum _{m=-\infty }^{\infty }{u}_m{A}_{n-m}=\sum _{m=-\infty }^{\infty }\sum _{j=-\infty }^{\infty }{u}_j{v}_{m-j}{w}_{n-m}$$

(52)

where we also have

$$B_n(t)=\sum _{m=-\infty }^{\infty }{u}_m{A}_{n-m}=\sum _{l+m=n}\sum _{k+j=l}{u}_k{v}_j{w}_m=\sum _{k+j+m=n}{u}_k{v}_j{w}_m$$

(53)

□

Then, the squared derivative term is

$$u_x^2\left(x,t\right)=\sum _{n=-\infty }^{\infty }{C}_n(t){\mathrm{e}}^{i{k}_{n}x},{ C}_n(t)=Conv[(k_n{u}_n{)}^2]$$

(54)

Proof. Coefficient

$C_n(t)$.

$${ C}_n(t)=Conv[\left(k_n{u}_n{)}^2\right]=\sum _{m=-\infty }^{\infty }{k_{m}u}_m{k_{n-m}u}_{n-m}$$

□

We finally return to the Hamiltonian:

$$H=i\sum _{n=-\infty }^{\infty }[-c_0{A}_n(t)-{\displaystyle \frac{\alpha }{3}}{B}_n(t)+\beta C_n(t)]{\delta }_n$$

(55)

Then

$$H(t)=i[-c_0{A}_0(t)-{\displaystyle \frac{\alpha }{3}}{B}_0(t)+\beta C_0(t)]=i{D}_0(t)$$

(56)

where

$$\begin{gathered} A_0(t)=\sum _{n=-\infty }^{\infty }{u}_n{u}_{-n} \\ {B}_0(t)=\sum _{l+m+n=0}[u_l,u_m,u_n] \\ {C}_0(t)=-\sum _{n=-\infty }^{\infty }{k}_n^2{u}_n{u}_{-n} \end{gathered}$$

(57)

This leads to the Hamiltonian in terms of the Fourier modes of (14), $u_n(t)$:

$$\boxed{H(t)=i{c}_0\sum _{n=-\infty }^{\infty }{u}_n{u}_{-n}-i{\displaystyle \frac{\alpha }{3}}\sum _{l+m+n=0}[u_l,u_m,u_n]-i\beta \sum _{n=-\infty }^{\infty }{k}_n^2{u}_n{u}_{-n}}$$

(58)

Here we have written the nonlinear term of the Hamiltonian as three-wave interactions (the square brackets), which is physically expected for the classical KdV equation. In Hamiltonian mechanics it is often necessary to write the dependences: $H(q,p,t)$. However, herein, the form $H(q,p)$ naturally occurs, as can be seen in (58). Note, however, that $u_n=u_n\left(t\right)$ and $u_{-n}=u_{-n}\left(t\right)$ both depend implicitly on time. The lack of a direct temporal dependence in the Hamiltonian is important, particularly for writing the Hamiltonian–Jacobi equation and finding its solution, as will be discussed in a sequel paper. Using the coefficients $u_n=u_n\left(t\right)$ and $u_{-n}=u_{-n}\left(t\right)$, the Hamiltonian becomes

$$H(t)=i{c}_0\sum _{n=-\infty }^{\infty }{u}_n\left(t\right)u_{-n}\left(t\right)-i{\displaystyle \frac{\alpha }{3}}\sum _{l+m+n=0}\left[u_l\right(t),u_m(t),u_n(t\left)\right]-i\beta \sum _{n=-\infty }^{\infty }{k}_n^2{u}_n\left(t\right)u_{-n}\left(t\right).$$

(59)

Now use the following relation equating three-wave interactions with a double convolution (see the Appendix A):

$$\sum _{l+m+n=0}[u_l,u_m,u_n]=\sum _{l+m+n=0}{u}_l{u}_m{u}_n=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{u}_m{u}_n{u}_{n-m}=\sum _{m=-\infty }^{\infty }{u}_m\sum _{n=-\infty }^{\infty }{u}_n{u}_{n-m}$$

(60)

This gives the full Hamiltonian in terms of the Fourier spatial modes:

$$\boxed{H(t)=i{c}_0\sum _{n=-\infty }^{\infty }{u}_n(t)u_{-n}(t)-i\beta \sum _{n=-\infty }^{\infty }{k}_n^2{u}_n(t)u_{-n}(t)-i{\displaystyle \frac{\alpha }{3}}\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{u}_m{u}_n{u}_{n-m}}$$

(61)

A simple change of variables gives

$$H(t)=i{c}_0\sum _{n=-\infty }^{\infty }{u}_n(t)u_{-n}(t)-i\beta \sum _{n=-\infty }^{\infty }{k}_n^2{u}_n\left(t\right)u_{-n}(t)-i{\displaystyle \frac{\alpha }{3}}\sum _{n=-\infty }^{\infty }{u}_n\sum _{m=-\infty }^{\infty }{u}_{m-n}{u}_m$$

(62)

Or

$$\boxed{H(t)=i\sum _{n=-\infty }^{\infty }[(c_0{k}_n-\beta k_n^3)q_n{p}_n-{\displaystyle \frac{\alpha }{3}}\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{u}_m{u}_n{u}_{n-m}]}$$

Now recall that

$$q_n(t)={\displaystyle \frac{u_n\left(t\right)}{k_n}}$$

$$p_n\left(t\right)=u_{-n}\left(t\right)$$

It follows that the generalized Hamiltonian for quasiperiodicity in both space and time has the form

$$\boxed{H(q,p)=i\sum _{\mathbf{n}\in {\mathbb{Z}}^N}^{\infty }{\hat{\omega }}_{\mathbf{n}}(\mathbf{n}·\mathbf{k},q_{\mathbf{n}},p_{\mathbf{n}})q_{\mathbf{n}}{p}_{\mathbf{n}}}$$

(63)

$$\boxed{{\hat{\omega }}_{\mathbf{n}}(\mathbf{n}·\mathbf{k},q_{\mathbf{n}},p_{\mathbf{n}})=c_0{k}_n-\beta k_n^3-{\displaystyle \frac{\alpha }{3}}{k}_n\left(\sum _{m=-\infty }^{\infty }{u}_{m-n}{u}_m/p_{\mathbf{n}}\right)}$$

(64)

Notice that the Hamiltonian has the same form as the linear problem, but now the dispersion relation is for full nonlinear dispersion of the KdV Equation (12). To arrive at the linear limit of the KdV equation, we simply set $\alpha =0$ in (64) to obtain the linear dispersion relation (16).

It is interesting to note that the above expression for the Hamiltonian (63) suggests that all nonlinear, integrable soliton equations can be written in this form, provided they are first-order in time. Thus, the Hamiltonian consists of a summation of terms with the nonlinear dispersion relation times the product of the generalized coordinates and momenta (63).

For numerical calculations, if we maintain nonlinearity but consider only one degree of freedom in the absence of interactions, we find the dispersion relation, as can be seen in Section 3, to be ${\omega }_0\simeq c_{0}k-\beta k^3+\gamma a^2,$ $\gamma =9c_0/16k{h}^4$ and $\gamma a^2=24\beta k^3{q}^2$.

A careful calculation for two degrees of freedom has a 2 × 2 Riemann matrix (Osborne, Chapter 16 [18]) and gives two frequencies:

$${\omega }_{01}\simeq c_0{k}_1-\beta k_1^3+{\gamma }_1{a}_1^2,{\gamma }_1=9c_o/16k_1{h}^4$$

$${\omega }_{02}\simeq c_0{k}_2-\beta k_2^3+{\gamma }_2{a}_2^2,{\gamma }_2=9c_o/16k_2{h}^4$$

The diagonal elements of the periodic matrix are given in terms of the Ursell number to second order.

$$B_{11}=-2ln{q}_1,q_1\simeq U_1(1-U_1^2),U_1={\displaystyle \frac{3a_1}{4h^3{k}_1^2}}.$$

$$B_{22}=-2ln{q}_2,q_2\simeq U_2\left(1-U_2^2\right),U_2={\displaystyle \frac{3a_2}{4h^3{k}_2^2}}$$

And the off-diagonal element of the period matrix is given by the following, to leading order:

$$B_{12}=-\mathrm{l}\mathrm{n} q_{12},{ q}_{12}\simeq {\left({\displaystyle \frac{k_1-k_2}{k_1+k_2}}\right)}^2+32k_1{k}_2(k_1^2{q}_1^2+k_2^2{q}_2^2)/{(k_1+k_2)}^4$$

For numerical computations it is important to note the squared amplitude corrections in both the frequencies (${\gamma }_1{a}_1^2$, ${\gamma }_2{a}_2^2$) and the diagonal elements ($U_1^2~a_1^2,{ U}_2^2~a_2^2$) of the period matrix. This is a remarkable departure from the periodic Fourier analysis of linear problems, where these amplitude corrections do not occur. Only in the quasiperiodic Fourier analysis of nonlinear problems do these nonlinear corrections happen. Thus, it would seem that we would need the Fourier amplitudes $a_1^2$ and $a_2^2$ before we can determine them! Of course, the natural solution is to estimate these amplitudes by a simple calculation and then make corrections iterating after the fact. This is one of the challenges of the quasiperiodic Fourier analysis of nonlinear problems! If we perform a numerical simulation, these computations are quite simple because the amplitudes are specified in advance. Only when we analyze time series do we need to compute these amplitudes in an iterative fashion. Details of these computations will be given in a future paper.

6. Matrix Mechanics for the KdV Equation

We take up here Heisenberg’s procedure [3], which we apply to the quasiperiodic Fourier series given by (34)–(36) using the methods given above. The approach requires that we write the quasiperiodic Fourier series for the generalized coordinates and momenta as a single summation, rather than as nested summations (Osborne, Chapter 32 [18]):

$$\begin{gathered} q_n(t)={\displaystyle \frac{1}{k_n}}\sum _{\left\{j\in \mathbb{Z}:n=I(\mathbf{n}·\mathbf{k})\right\}}{u}_j{e}^{-i{\mathsf{\Omega }}_{j}t+i{\mathsf{\Phi }}_j} \\ {p}_n(t)=\sum _{\left\{j\in \mathbb{Z}:n=-I(\mathbf{n}·\mathbf{k})\right\}}{u}_j{e}^{-i{\mathsf{\Omega }}_{j}t+i{\mathsf{\Phi }}_j} \end{gathered}$$

(65)

Here the index, j, is the ordering parameter used in Table 1 and Figure 4. This enforces the limits $-J\le j\le J$ on the above series. Furthermore, we compute $J=\left[\right(2M+1{)}^N+1]/2$, where M is the limit of the nested summations and N is the number of nested sums. We use the definition

$$I\left(\mathbf{n}·\mathbf{k}\right)={\displaystyle \frac{L}{2\pi }}\sum _{l=1}^N{n}_l{k}_l$$

(66)

which fixes the meaning of the above summations. The frequencies and phases used above are defined by

$$\begin{gathered} {\mathsf{\Omega }}_j=\sum _{l=1}^N{n}_l^j{\omega }_l \\ {\mathsf{\Phi }}_j=\sum _{l=1}^N{n}_l^j{\varphi }_l \end{gathered}$$

(67)

The coordinates and momenta become

$$\begin{gathered} q_n(t)=\sum _{\left\{j\in \mathbb{Z}:n=I(\mathbf{n}·\mathbf{k})\right\}}{q}_j^n{e}^{-i{\mathsf{\Omega }}_{j}t+i{\mathsf{\Phi }}_j} \\ {p}_n(t)=\sum _{\left\{j\in \mathbb{Z}:n=-I(\mathbf{n}·\mathbf{k})\right\}}{p}_j^n{e}^{-i{\mathsf{\Omega }}_{j}t+i{\mathsf{\Phi }}_j} \end{gathered}$$

(68)

We then make the following choice

$$\begin{gathered} q_j^n={\displaystyle \frac{u_{{\mathbf{n}}^j}{\delta }_{n,I({\mathbf{n}}^j·\mathbf{k})}}{k_n}}{e}^{i{\mathsf{\Phi }}_j} \\ {p}_j^n=u_{{\mathbf{n}}^j}{\delta }_{n,I({\mathbf{n}}^j·\mathbf{k})}{e}^{i{\mathsf{\Phi }}_j} \end{gathered}$$

(69)

where $\mathsf{\delta }$ is the Kronecker delta. Finally, we have the simple expressions for the generalized coordinates and momenta:

$$\begin{gathered} q_n(t)=\sum _{j=-J}^J{q}_j^n{e}^{-i{\mathsf{\Omega }}_{j}t} \\ {p}_n(t)=\sum _{j=-J}^J{p}_j^n{e}^{-i{\mathsf{\Omega }}_{j}t} \end{gathered}$$

(70)

At this point Heisenberg [3] suggests that we write the coordinates in matrix form (see also Weinberg [24]):

$${\mathbf{q}}_n(t)\approx \left[\begin{array}{ccccccc}\ddots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \\ \cdots & q_0^n& q_1^n{e}^{-i{\mathsf{\Omega }}_{1}t}& q_2^n{e}^{-i{\mathsf{\Omega }}_{2}t}& q_3^n{e}^{-i{\mathsf{\Omega }}_{3}t}& q_4^n{e}^{-i{\mathsf{\Omega }}_{4}t}& \cdots \\ \cdots & q_{-1}^n{e}^{-i{\mathsf{\Omega }}_{-1}t}& q_0^n& q_1^n{e}^{-i{\mathsf{\Omega }}_{1}t}& q_2^n{e}^{-i{\mathsf{\Omega }}_{2}t}& q_3^n{e}^{-i{\mathsf{\Omega }}_{3}t}& \cdots \\ \cdots & q_{-2}^n{e}^{-i{\mathsf{\Omega }}_{-2}t}& q_{-1}^n{e}^{-i{\mathsf{\Omega }}_{-1}t}& q_0^n& q_1^n{e}^{-i{\mathsf{\Omega }}_{1}t}& q_2^n{e}^{-i{\mathsf{\Omega }}_{2}t}& \cdots \\ \cdots & q_{-3}^n{e}^{-i{\mathsf{\Omega }}_{-3}t}& q_{-2}^n{e}^{-i{\mathsf{\Omega }}_{-2}t}& q_{-1}^n{e}^{-i{\mathsf{\Omega }}_{-1}t}& q_0^n& q_1^n{e}^{-i{\mathsf{\Omega }}_{1}t}& \cdots \\ \cdots & q_{-4}^n{e}^{-i{\mathsf{\Omega }}_{-4}t}& q_{-3}^n{e}^{-i{\mathsf{\Omega }}_{-3}t}& q_{-2}^n{e}^{-i{\mathsf{\Omega }}_{-2}t}& q_{-1}^n{e}^{-i{\mathsf{\Omega }}_{-1}t}& q_0^n& \cdots \\ \cdots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right]$$

(71)

Here we have used Heisenberg’s convention such that each row of the matrix contains sequential terms in the quasiperiodic Fourier series, $q_n\left(t\right)$. Furthermore, each row is also shifted by one element so that the vacuum term always occurs on the matrix diagonal. It is further convenient to write the elements of the Heisenberg matrix as $q(m,n)$. Then with this notation we have

$$\begin{gathered} q_n(k,l,t)=q_{k-l}{e}^{i({\mathsf{\Omega }}_k-{\mathsf{\Omega }}_l)t}=q_{k-l}{e}^{i(k-l)\mathsf{\Omega }t} \\ {p}_n(k,l,t)=p_{k-l}{e}^{i({\mathsf{\Omega }}_k-{\mathsf{\Omega }}_l)t}=p_{k-l}{e}^{i(k-l)\mathsf{\Omega }t} \end{gathered}$$

(72)

The definition of the product of the two matrices has the form

$$\{{\mathbf{q}}_n(k,l,t)\}\{{\mathbf{p}}_n(k,l,t)\}=\sum _{i=-\infty }^{\infty }{q}_n(k,i,t)p_n(i,l,t)$$

(73)

At this point it is important to notice that the frequency, $\mathsf{\Omega }(k,l)$, associated with each coefficient, $q(k,l)$, can then be written using the following expression:

$$\mathsf{\Omega }(k,i)+\mathsf{\Omega }(i,l)=\mathsf{\Omega }(k,l)$$

(74)

This suggests that we must have a large array of incommensurable frequencies of the form

$${\mathsf{\Omega }}_{-J}\dots {\mathsf{\Omega }}_{-2},{\mathsf{\Omega }}_{-1},{\mathsf{\Omega }}_0,{\mathsf{\Omega }}_1,{\mathsf{\Omega }}_2\dots {\mathsf{\Omega }}_J$$

Also note that

$$\mathsf{\Omega }(k,l)={\mathsf{\Omega }}_k-{\mathsf{\Omega }}_l$$

(75)

This is because we are dealing with frequencies that are incommensurable, such that the coordinates and momenta are quasiperiodic Fourier series. We also note that $q(k,l)$ are complex conjugates of $q(l,k)$.

We then find the following generalized coordinate matrix:

$${\mathbf{q}}_n\left(t\right)=\left[\begin{array}{ccccccc}\ddots & ⋮& ⋮& ⋮& ⋮& ⋮& ·\\ \cdots & q\left(\mathrm{0,0}\right)& q\left(\mathrm{0,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,1}\right)t}& q\left(\mathrm{0,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,2}\right)t}& q\left(\mathrm{0,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,3}\right)t}& q\left(\mathrm{0,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,4}\right)t}& \cdots \\ \cdots & q\left(\mathrm{1,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,0}\right)t}& q\left(\mathrm{1,1}\right)& q\left(\mathrm{1,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,2}\right)t}& q\left(\mathrm{1,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,3}\right)t}& q\left(\mathrm{1,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,4}\right)t}& \cdots \\ \cdots & q\left(\mathrm{2,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,0}\right)t}& q\left(\mathrm{2,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,1}\right)t}& q\left(\mathrm{2,2}\right)& q\left(\mathrm{2,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,3}\right)t}& q\left(\mathrm{2,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,4}\right)t}& \cdots \\ \cdots & q\left(\mathrm{3,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,0}\right)t}& q\left(\mathrm{3,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,1}\right)t}& q\left(\mathrm{3,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,2}\right)t}& q\left(\mathrm{3,3}\right)& q\left(\mathrm{3,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,4}\right)t}& \cdots \\ \cdots & q\left(\mathrm{4,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,0}\right)t}& q\left(\mathrm{4,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,1}\right)t}& q\left(\mathrm{4,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,2}\right)t}& q\left(\mathrm{4,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,3}\right)t}& q\left(\mathrm{4,4}\right)& \cdots \\ \cdots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right]$$

(76)

The matrix of the momenta may also be written as

$${\mathbf{p}}_n\left(t\right)=\left[\begin{array}{ccccccc}\ddots & ⋮& ⋮& ⋮& ⋮& ⋮& ·\\ \cdots & p\left(\mathrm{0,0}\right)& p\left(\mathrm{0,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,1}\right)t}& p\left(\mathrm{0,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,2}\right)t}& p\left(\mathrm{0,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,3}\right)t}& p\left(\mathrm{0,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{0,4}\right)t}& \cdots \\ \cdots & p\left(\mathrm{1,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,0}\right)t}& p\left(\mathrm{1,1}\right)& p\left(\mathrm{1,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,2}\right)t}& p\left(\mathrm{1,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,3}\right)t}& p\left(\mathrm{1,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{1,4}\right)t}& \cdots \\ \cdots & p\left(\mathrm{2,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,0}\right)t}& p\left(\mathrm{2,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,1}\right)t}& p\left(\mathrm{2,2}\right)& p\left(\mathrm{2,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,3}\right)t}& p\left(\mathrm{2,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{2,4}\right)t}& \cdots \\ \cdots & p\left(\mathrm{3,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,0}\right)t}& p\left(\mathrm{3,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,1}\right)t}& p\left(\mathrm{3,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,2}\right)t}& p\left(\mathrm{3,3}\right)& p\left(\mathrm{3,4}\right)e^{-i\mathsf{\Omega }\left(\mathrm{3,4}\right)t}& \cdots \\ \cdots & p\left(\mathrm{4,0}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,0}\right)t}& p\left(\mathrm{4,1}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,1}\right)t}& p\left(\mathrm{4,2}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,2}\right)t}& p\left(\mathrm{4,3}\right)e^{-i\mathsf{\Omega }\left(\mathrm{4,3}\right)t}& p\left(\mathrm{4,4}\right)& \cdots \\ \cdots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right]$$

(77)

It then follows that the commutator for q and p is given by

$$\mathbf{q}\mathbf{p}-\mathbf{p}\mathbf{q}=i{\displaystyle \frac{h}{2\pi }}\mathbf{I}$$

(78)

where, with Heisenberg, we have introduced the Planck constant, h, and the identity, I. This is of course one of the most fundamental relations of quantum mechanics, i.e., the noncommutativity of the coordinates and momenta in terms of Planck’s constant. It is well known that the latter expression is related to the uncertainty principle.

Heisenberg has made it clear that the matrices for the coordinates and momenta have the form

$$\begin{gathered} q(k,l)=q(k,l)e^{-i\mathsf{\Omega }(k,l)t} \\ p(k,l)=p(k,l)e^{-i\mathsf{\Omega }(k,l)t} \end{gathered}$$

(79)

These are the fundamental matrices associated with the dynamics of the KdV equation, which are important for better understanding the quantum mechanics of the soliton equation in terms of finite gap theory.

7. Properties of the Spatially Periodic Fourier Modes, $u_n\left(t\right)$

The Hamiltonian is a constant for evolution described by the KdV equation. Time is not explicit in the Hamiltonian but is implicit in the qs and ps ((63) and (64)). We can see that the three-wave interactions are derived from the nonlinear term (proportional to $\alpha$) in the KdV equation. The time evolution of the spatially periodic Fourier modes, $u_n\left(t\right)$, is given by quasiperiodic Fourier series (19). We now look at some properties of these latter modes.

• Positive Fourier wavenumber components:

The Fourier coefficients in (14) are now written in the following, slightly different, notation:

$$\boxed{u_n(t)=\sum _{\{\mathbf{n}\in {\mathbb{Z}}^N: n=\mathbf{n}·\hat{\mathbf{n}}\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}}$$

(80)

The summation is over the integer vector $\mathbf{n}\in {\mathbb{Z}}^N$, where N is the genus (the number of nonlinear modes) and the Riemann matrix is $N\times N$. The vector $\hat{\mathbf{n}}=L\mathbf{k}/2\pi =[1, 2, 3...N]$ consists of the natural numbers, $\mathbb{N}$. The dot product, $\mathbf{n}·\hat{\mathbf{n}}$, lies on $\mathbb{Z}$, $-\infty ,\dots -3,-2,-1, 0, 1, 2, 3...\infty$. Therefore, n also lies on the set of all integers, $\mathbb{Z}$. Because of the structure of the quasiperiodic Fourier series (80), each value of n has an infinite number of terms to sum; this explains the meaning of the summation notation. The generalized coordinates and momenta are

$$q_n=u_n/k_n, u_n=k_n{q}_n,\mathrm{for} n>0$$

Then the generalized coordinates are

$$\boxed{q_n(t)={\displaystyle \frac{1}{k_n}}\sum _{\{\mathbf{n}\in {\mathbb{Z}}^N: n=\mathbf{n}·\hat{\mathbf{n}}>0\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}}$$

(81)

Note that only the particular terms in (81) for which $\mathbf{n}·\hat{\mathbf{n}}>0$ contribute to $q_n\left(t\right)$.

• Zero or ground-state Fourier wavenumber components:

The zeroth elements are the “vacuum states”,$u_0\left(t\right),q_0\left(t\right)$: This is the “dynamical ground state”, which has a time varying background given by

$$\boxed{u_0(t)=\sum _{\{\mathit{n}\in {\mathbb{Z}}^N: n=\mathbf{n}·\hat{\mathbf{n}}=0\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathit{\omega }t+i\mathbf{n}·\mathsf{\varphi }}}$$

(82)

We should recognize that $u_0\left(t\right)$ is formally the mean value of $u(x,t)$. However, according to the above equation, the mean of this system is generally never zero, even in the classical case, except possibly at $t=0$. In both the classical and quantum cases, the ground state has time dynamics (82) in which nonlinear modes continually and spontaneously jump up out of the vacuum state.

Let us study some of the physical behavior of the vacuum state. We first show how this state can never remain at zero, even if we arbitrarily set it to zero at time zero: $u_0(t=0)=0$. This happens because we first assume that the wavenumbers are commensurable, but the frequencies are incommensurable, and the frequency harmonics can never be zero because they are computed by the nonlinear dispersion relation (64). Furthermore, the harmonics generally can never be integers but must lie only on the discretuum.

The harmonic frequency is $\mathbf{n}·\mathsf{\omega }=n_1{\omega }_1+n_2{\omega }_2+\dots$, which results from $\mathbf{n}·\mathbf{k}=n_1{k}_1+n_2{k}_2+\dots =0, k_1=k, k_2=2k$. Let us consider the case for which $\mathbf{n}=[n_1,n_2]=[2,-1]$. Then,

$$\mathbf{n}·\mathbf{k}=n_1{k}_1+n_2{k}_2=0$$

(83)

But

$$\mathbf{n}·\mathsf{\omega }=n_1{\omega }_1+n_2{\omega }_2=2{\omega }_1-{\omega }_2={\mathsf{\Omega }}_{\mathbf{n}}\ne 0$$

(84)

We notice that, generally speaking:

$$\begin{gathered} K_{\mathbf{n}}=\mathbf{n}·\mathbf{k}=0 \\ {\mathsf{\Omega }}_{\mathbf{n}}=\mathbf{n}·\mathsf{\omega }={\delta }_{\mathbf{n}}<<1 \end{gathered}$$

(85)

These are the conditions that occur in nonlinear water wave resonances for zero harmonic wavenumbers and small-frequency harmonics. In the case of three-wave interactions, (shallow-water waves for the KdV equation) we have

$$\begin{gathered} k_1+k_2+k_3=0 \\ {\omega }_1+{\omega }_2+{\omega }_3\simeq 0 \end{gathered}$$

(86)

In the case of four-wave interactions, we have (in familiar notation for deep water, in terms of envelope equations, such as the nonlinear Schrödinger, Dysthe, Trulsen–Dysthe and Zakharov equations):

$$\begin{gathered} k_1+k_2+k_3+k_4=0 \\ {\omega }_1+{\omega }_2+{\omega }_3+{\omega }_4\simeq 0 \end{gathered}$$

(87)

This reminds us that the quasiperiodic Fourier series solutions of KdV (20) can themselves be written as follows:

$$\begin{gathered} u(x,t)=\sum _{n=1}^{\infty }{u}_n\mathrm{c}\mathrm{o}\mathrm{s}(k_{n}x-{\omega }_{n}t+{\varphi }_n)+ \\ +\sum _{m=1}^{\infty }\sum _{n>m}^{\infty }{u}_{mn}\mathrm{c}\mathrm{o}\mathrm{s}[(k_m\pm k_n)x-({\omega }_m\pm {\omega }_n)t+({\varphi }_m\pm {\varphi }_n)]+ \\ +\sum _{m=1}^{\infty }\sum _{n>m}^{\infty }\sum _o^{\infty }{u}_{mno}\mathrm{c}\mathrm{o}\mathrm{s}\left[\left(k_m\pm k_n\pm k_o\right)x-\left({\omega }_m\pm {\omega }_n\pm {\omega }_o\right)t+\left({\varphi }_m\pm {\varphi }_n\pm {\varphi }_0\right)\right]+ \\ +\sum _{m=1}^{\infty }\sum _{n>m}^{\infty }\sum _o^{\infty }\sum _p^{\infty }{u}_{mnop}\mathrm{c}\mathrm{o}\mathrm{s}[(k_m\pm k_n\pm k_o\pm k_p)x-({\omega }_m\pm {\omega }_n\pm {\omega }_o\pm {\omega }_p)t+({\varphi }_m\pm {\varphi }_n\pm {\varphi }_o\pm {\varphi }_p)] \end{gathered}$$

(88)

The terms with three-wave resonant interactions (86) dominate in shallow water, while the four-wave resonant interactions (87) dominate in deep water for the classical nonlinear wave problem. These various behaviors occur for the different soliton equations and are responsible for the many types of coherent structures, such as solitons, Stokes waves, breathers, superbreathers, kinks, vortices, etc., all of which must also occur in quantum mechanics. The goal of this paper is to consider the possibility that classical nonlinear integrability and its properties can be used to study the corresponding quantum integrability using finite gap theory for wave mechanics, following the paths of Heisenberg and Schrödinger to microscopic scales.

An interesting thought is that if a small particle with a mass $~{10}^{-18}$ times that of the electron, say, were to be discovered in the future, one might be able to visualize quantum scales down to or near to the Planck scale. The quasiperiodic Fourier series of this paper might then be useful for analyzing hypothetical measurements of this type. We view this observation as the result of an interesting thought experiment about the nature of quantum mechanics, not necessarily as a real possibility.

The above relation (82) for $u_0\left(t\right)$ describes the vacuum state and the resonance conditions. It has a natural (slow) time evolution and thus can never be zero during the space–time evolution of the KdV equation, even if we set it to zero at zero time. This relates to the familiar quantum mechanical result such that the ground state has some finite value, never zero. It also says that even in the vacuum state we get the ubiquitous occurrence of the nonlinear modes (at long periods and frequencies) that are derived from the “particle-like soliton” modes on the diagonal of the Riemann matrix. The random appearing modes seem to be a kind of classical analog of the ground state quantum modes which jump up randomly out of the vacuum state, assuming random phases in the requisite quasiperiodic Fourier series.

• Negative Fourier wavenumber components:

The negative Fourier coefficients in (14) have the form

$$\boxed{u_{-n}(t)=\sum _{\{\mathbf{n}\in {\mathbb{Z}}^N:-n=\mathbf{n}·\hat{\mathbf{n}}<0\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}}$$

(89)

where the momenta are defined by

$$p_n=u_{-n}$$

Then the momenta, as quasiperiodic Fourier series, are given by

$$\boxed{p_n(t)=\sum _{\{\mathbf{n}\in {\mathbb{Z}}^N:-n=\mathbf{n}·\hat{\mathbf{n}}<0\}}{u}_{\mathbf{n}}{e}^{-i\mathbf{n}·\mathsf{\omega }t+i\mathbf{n}·\mathsf{\varphi }}}$$

(90)

We can see that only when $\mathbf{n}·\hat{\mathbf{n}}<0$ do the summations contribute to the generalized momenta, $p_n\left(t\right)$.

8. Summary of Results in This Paper on the Classical and Quantum Problems for the KdV Equation, and a Few Future Problems

(1)

We first discussed how the solution to the KdV equation, in terms of the Its–Matveev formula, could be expanded in terms of a quasiperiodic Fourier series (7) whose coefficients are written in terms of the Riemann matrix by (9)–(11) with nonlinear dispersion (12): We refer to this operation, which uses the Baker-Mumford theorem [25,26], as a kind of inverse convolution. This theorem forms the most general single valued, multiply periodic, meromorphic functions which are written in terms of Riemann theta functions. Functions of this type solve soliton equations and there are only three fundamental ways to do this. Small variations in these three ways allow for the solution of an infinity of soliton equations. Furthermore, in soliton solutions of this type, the Riemann matrix itself is symmetric. This means that, as the result of this symmetry, the KdV soliton equation has as its spectral solutions a linear superposition law which is a quasiperiodic Fourier series, a convenient starting place for the application of integrable soliton nonlinearity to quantum mechanics. Some might find the theory just discussed as too complicated to deal with on a daily basis. However, one has only to look at Chapter 32 of Osborne [18] to see that these methods are the “bread and butter” of the study of nonlinear soliton interactions in water waves from theoretical, numerical and experimental points of view. The algorithms are found to be “hyperfast” with respect to conventional numerical methods. Applying these methods to quantum mechanics can only provide additional tools for the study of integrable nonlinearity at micro scales.

(2)

The space–time quasiperiodic Fourier series spectral solutions of KdV (7) are then rewritten as being periodic in space and quasiperiodic in time (14) and (19) with nonlinear dispersion (18). It then follows naturally that the time-varying Fourier coefficients (19) solve the set of ODEs for the Fourier coefficients, $u_n\left(t\right)$ (15). This result was found by first solving the KdV equation with quasiperiodic Fourier series (7), in both space and time, and then reducing this result to periodicity in space and quasiperiodicity in time. Our results are therefore consistent with the assumptions first used to study the Fermi–Pasta–Ulam problem (FPU recurrence) [8], in the discovery of the soliton by Zabusky and Kruskal [9] and in the discovery of the inverse scattering transform [10]. The study of further details of the quantum FPU problem is a project for a future study.

(3)

We then put the KdV equation into Hamiltonian form based upon the results of Gardner [16] and Faddeev and Takhtajan [11]. We discuss how Hamilton’s Equations (30) and (31) (written as a function of $u_n\left(t\right)$) are solved by the quasiperiodic Fourier series for $u_n\left(t\right)$ (19).

(4)

Formally, the Hamiltonian can also be written in terms of the q’s and p’s, such that the quasiperiodic Fourier series solutions are given by (33). This discovery is a consequence of the symmetries discussed above in the KdV equation. We note that Equation (33) expresses the conditions assumed by Heisenberg to develop matrix mechanics (on SU(2)). While in the present case we are dealing with classical fields, the results nevertheless are equivalent to the formulation for particles.

(5)

Furthermore, we have shown that the fundamental generalized coordinates and momenta for the quantum mechanical KdV problem can be computed in terms of the finite gap spectrum (the Riemann matrix, wavenumbers, frequencies and phases) of the quasiperiodic inverse scattering transform. This means that all parameters of the classical KdV equation are present in the quantum matrix mechanics, most particularly in terms of the Riemann matrix. Since the diagonal elements of this matrix correspond to solitons, in the quantum mechanical problem we deal directly with solitons and their interactions (the off-diagonal elements). A future problem deals with the construction of the Hamiltonian–Jacobi equation and with it the derivation of the associated Schrödinger equation.

(6)

The ground state of the KdV equation is found to never be zero, but instead a time-varying source of KdV modes determined from the Riemann spectrum. These modes continually jump up out of the vacuum state both for the classical and quantum problems. These modes are the quantum analogue of classical infra gravity waves.

(7)

We refer to the problem studied here as the quantum Fermi–Pasta–Ulam problem (qFPU). A future paper will (1) clarify the role of solitons and their nonlinear interactions in the quantum problem and (2) address the associated Schrödinger equation with soliton solutions, all derived directly from the parameters of the finite gap theory of soliton equations with quasiperiodic boundary conditions. We emphasize that the integrability of the classical KdV equation also implies the complete integrability of the quantum problem.

(8)

The soliton modes in the quasiperiodic Fourier series are not easily computed in numerical modeling efforts. Two steps are normally used: (a) a modular transformation to map the oscillatory modes (waves) to solitons (particles); (b) the theta function is more naturally written as a Gaussian series in the soliton limit [12,18]. Both of these methods save huge amounts of computer time in numerical models.

(9)

The issue of the numerical modeling of the KdV has been partially studied in Chapter 32 of reference [18], where the Its–Matveev formula (7) was implemented. However, the associated quasiperiodic Fourier series in the right-hand part of (7) has yet to be numerically implemented and remains a project for the future. The complexities of the quasiperiodic Fourier series provide significant challenges for modeling development, but they are not insurmountable. Modeling the physics of the associated quantum problem will follow using the Schrödinger equation.

(10)

It is further relevant to discuss how these results provide a pathway to computing classical/quantum nonlinear wave dynamics on a quantum computer. Two issues are important in this regard. First, a quantum algorithm (Q) needs to be developed for the quasiperiodic Fourier transform (qFFT), which we here refer to as the QqFFT, in complete analogy with the quantum (periodic) FFT discussed by [27]. Second, the mathematics of matrix mechanics (discussed in Section 6, above) leads to natural ways to program quantum computers (see also Nielsen and Chuang [27]). A future paper will develop the QqFFT algorithm and explore its application to quantum computing. We are of course anticipating the development of quantum computers in the near future.