Introduction

The Kortweg-de Vries (KdV) equation is a nonlinear partial differential equation of third order, as follows:

First formulated as part of an analysis of shallow-water waves in canals, it has subsequently been found to be involved in a wide range of physics phenomena, especially those exhibiting shock waves, traveling waves, and solitons. Certain theoretical physics phenomena in the quantum mechanics domain are explained by means of a KdV model. It is used in fluid dynamics, aerodynamics, and continuum mechanics as a model for shock wave formation, solitons, turbulence, boundary layer behavior, and mass transport. It has been studied and applied for many decades. Many different closed-form, series approximation, and numerical solutions are known for particular sets of boundary and initial conditions. Top

In the following, we shall be applying commutative hypercomplex mathematics. This is a system of mathematics that obeys the axioms of the classical complex variables and behaves in all ways like the classical complex analysis, while treating a 4-D independent variable. In order to understand the following, the reader should first review the Hypercomplex Math page.

We shall present something new in the form of the general analytical solution for the KdV equation in terms of commutative hypercomplex mathematics. We shall show that, unlike for linear partial differential equations, there is not a variety of eigenfunctions from which a solution may be constructed in a variety of ways, but only one characteristic function that satisfies the equation when analyticity is required. The solution will be presented as an analytic function of one 4-D variable and as a complementary pair of functions of a classical complex variable. The latter are further reduced to a complementary pair of real-valued functions that are verified as solutions of the Korteweg equation. Top

We emphasize analytical solution because the solution that we shall develop is analytic in the classical complex variable sense (i.e., is continuous and single-valued in the region of interest) and is "conservative" in physics terminology. Other solutions might be available that do not preserve, for example, total energy or mass in the full four-space. The solution is general in that we shall first transform the PDE into an ODE and then perform three integrations to obtain a solution with three arbitrary constants of integration. It is also general in the sense that it is a 4-D function that implicitly contains several different solution forms in 2-D and in one dimension and time, as we shall show.

One might recall that partial differential equations, especially those used to model the behavior of some material substance, typically describe the behavior of some variable, parameter, or physical effect about a point. They are typically derived from a force-energy-momentum-mass balance on an infinitesimal element at a point. As an aside, we note that they usually represent approximations only to first or second order of the effects that they describe, in order to keep the equations tractable. We might obtain a closed-form solution for the PDE, only to find that it does not accurately describe the extremes of the physical behavior that we are attempting to model. Top

If we obtain a characteristic function of the KdV equation by whatever means, then the function will embody the same description of physical effects as the PDE and must be viewed and applied in the same way; i.e., as describing the variation of effects about a point, and not necessarily the macro behavior over all space. The KdV equation was developed to describe the immediate, localized reaction of a tiny, incremental element of mass to given external forces and momentum and energy inputs. In physics terms, the integrated result is expressed in body-centered coordinates whose origin moves with the subject particle of mass. At any given instant of time and for given local conditions, the integrated result indicates how the particle of mass will move next within the body-centered frame. We continue to emphasize: The integrated result describes an immediate, localized reaction, and says nothing about the long-term motion of a given particle of mass. For that reason, we must do a finite-element-like numerical calculation in order to coordinate the motions and interactions of all the particles, thereby obtaining a view of the overall motion of the fluid. This view explains why there is not any analytic-function solution of the KdV equation that models turbulent behavior in the large. Any "solution" is point-localized. Top

Hypercomplex Solution

As mentioned earlier, our basic approach to solution is to first convert the (nonlinear) KdV equation to an ODE, then solve it by means of classical methods. To convert partial differentials to ordinary derivatives, we shall use the following consequences of the 4-D Cauchy-Riemann equations: Top

where Z=1x+iy+ jz+kct. Refer to the Hypercomplex Math page for a derivation. These relations transform partial differentials into an ordinary derivative with respect to a 4-D independent variable, Z. Observe that the first two equalities on the left are identical to those for 2-D classical complex variables, and the two on the right are extensions for the 4-D case. In the rightmost relation, c is the characteristic speed of propagation in whatever medium we are working. These relations, being direct consequences of the 4-D Cauchy-Riemann conditions, are therefore continuity conditions. By applying them to the KdV equation, we force continuity upon the solution. One might argue that we are simultaneously solving the KdV equation with the Cauchy-Riemann equations. However, that would be somewhat misleading because the Cauchy-Riemann equations are satisfied by any analytic function.

Making the partial derivative conversions, we obtain:

where Z is a 4-D variable, k is a 4-D algebraic basis element, and c is the characteristic speed of propagation of effects in a KdV medium. This equation is still nonlinear, but is solvable by simple, direct methods. A first integration yields: Top

where A is an arbitrary 4-D constant of integration. This is again integratable. We rearrange, multiply both sides of the equation by the first derivative, then integrate to get: Top

The element B is another 4-D constant of integration. This equation is integratable if we can effect a factorization of the polynomial in u [see W. F. Ames, Nonlinear ordinary differential equations in transport processes (Academic Press, New York, 1968), p. 55]. From simple theory of polynomials, there will be three factors and they will all be arbitrary, because they will all be functions of the arbitrary constants A,B and simple constants. Let us denote them by a,b,e; then we have Top

From this, we see that an unavoidable condition -(a+b+e)=kc/2 has been imposed on the allowable values of the factors, because kc/2 is a constant. This effectively reduces the number of independent factors to two. For convenience, we shall designate a and b as the independent, arbitrary factors; then e=-kc/2-(a+b). This represents no loss of generality, because at this point we have only two arbitrary constants of integration. Top

Indeed, there is no good reason to explicitly solve for the factors in terms of the (arbitrary) constants of integration. Instead, we can simply use the factored notation and the two independent factor symbols as surrogates for the (so far) two arbitrary constants of integration. With that insight, we can use Ames' solution for the third and final integration, yielding: Top

The third and final, 4-D constant of integration is C. The function sn[...] is the Jacobi elliptic function, in this case with a 4-D argument, Z. This solution for the ODE-form KdV equation is easily verified in Mathematica, treating Z as the independent variable and treating all constants in the conventional way as regards differentiation. This result is a rather busy-looking formula whose form we can simplify and standardize by defining more conventional-looking parameters, to wit:

All elements on the right-hand sides of these definitions are either constant or arbitrary, so the newly-defined parameters are arbitrary. Note the difference in the new, italicized parameter k and the unitalicized, algebraic basis element k. With these definitions, the solution is: Top

This is the general analytical solution for the KdV equation in terms of the 4-D commutative hypercomplex variable Z. That means that it is the complete solution for the ODE form inasmuch as we have integrated three times and have a solution including three arbitrary constants of integration. Moreover, it is the complete solution (i.e., the most general characteristic function) of the PDE form of the KdV equation when continuity conditions are imposed. Again, I remark: Continuity conditions are nearly always assumed for PDEs, but are seldom explicitly imposed. Top

All three constants in the solution are arbitrary and are derived indirectly from the constants of integration, as indicated. They can be real, classical complex, or 4-D hypercomplex, and can be manipulated to satisfy boundary and/or initial conditions. In all cases, we have a means of interpretation, as we shall show.

In the u(Z) expression, the ± sign merely indicates that either positive or negative-going waves are allowed, so for convenience we shall discuss only the positive case from this point forward. Also, the i sqrt(2) factor shall be assumed to be subsumed into the arbitrary constants, and will not be further shown. Top

We remind the reader that this is not the one and only, final solution for all problems in a KdV medium. Just as does the KdV PDE, it merely represents the behavior about a point. Given an external force, impulse, etc. on a small increment of material at the point, the characteristic function that we have found indicates how the disturbance will be propagated away from that point in the absence of any other disturbances, reflections, interferences, etc. in the medium. Of course, in a real problem all parts of the medium might be simultaneously receiving independent impulses, and the net external impulse on a specific increment of material at a given time is an integration over all the disturbances originating in all of the rest of the material increments, taking account of the fact that the speed of propagation might vary from point to point in the medium. Consequently, a solution for a given problem involving arbitrary boundary and initial conditions must be pieced together point by point within each small time increment, similar to what is done in a finite element method. It might seem that we have gained no advantage. However, the analytical solution can yield further theoretical insight and provide new avenues for numerical solution. Top

The above solution is in terms of one 4-D independent variable, Z. We can break this down into two separate solutions in terms of one classical complex variable, each, by writing it in the canonical form, as follows. Recall the definition of a commutative hypercomplex function or operator: Top

The eigenvalues are classical complex variables, and the above definition means that a differential operator that operates on the 4-D variable Z also operates separately on the two eigenvalues of Z, and that the 4-D solution u (Z) is the sum of two 2-D solutions u () and   u ():

i.e., the eigenfunctions of u (Z) in the canonical form are also solutions of the original PDE. Top

When we are given a 4-D expression such as u (Z) and we wish to expand it into canonical form, it is not so simple as writing two identical copies with just different independent variables. We must write every component of the u (Z) expression, including all constants, in canonical form and then reduce the resulting expression into simplest canonical form, as follows. We first expand the arbitrary 4-D constants:

All of the subscripted components on the right (excluding the e-basis vectors) are real or classical complex numbers. Top

Recall that any 4-D commutative hypercomplex entity, such as an analytic function of a 4-D variable, can be represented in 4 X 4 real matrix form. If this were done for our u (Z) solution, then the eigenfunction components and their complex conjugates would be the eigenvalues of the matrix form of u (Z). Again, each of the eigenfunctions of u (Z) is a solution of a KdV equation which is stated in terms of the associated 2-D independent variable or . Top

In the above notation, selecting the (+) sign in front, the eigenfunctions are:

All terms are either real or classical complex. We can carry this a step further and expand u (Z) into 4-D vector form:

This time, the individual vector components of u (Z) definitely are not solutions of a KdV equation, because the KdV equation is nonlinear. Top

Elementary Properties

We have obtained a solution u (Z) in terms of one variable having three space dimensions and time. One of the first questions that must be answered is, "Does it reduce to a solution of the original, one-dimensional KdV PDE when the y,z components are set to zero?" This is easy enough to check. Setting y=z=0, we get: Top

Now, because we have established a classical analysis that is applied to 4-D functions (see the Hypercomplex Math page), we can treat the 4-D constants and the algebraic basis element k as simple constants with respect to differentiation, much as we would the classical imaginary, i, taking account that k²=1 and k^-1=k. We leave it to the reader to put the above result into Mathematica and verify that it is a solution of the KdV PDE. Top

Our reduced, one-dimensional solution u(x,ct), above, is not immediately illuminating about what we know of the physical situation that it is used to model. We can discern the true behavior when we examine the eigenfunctions u₁(), u₂() with the y,z coordinates set to zero: Top

As before, all of the subscripted constants on the right are arbitrary real or complex quantities. We can choose meaningful real values for the constants such that we have purely real expressions that are each solutions of the KdV PDE. The sn(x,k) function (below) has the qualitative behavior of the ordinary sin(x) function, except that it has more-rounded peaks and period 4K(k), where K(k) is the complete elliptic integral of the first kind. Top

Consequently, the sn^-2(x,k) function represents a periodic wavetrain with the following qualitative behavior and period 2K(k):

The periodic spikes go to infinity, which is something that is never seen in real materials, such as in water. In the real eigenfunctions, above, the minimum amplitude is set by the square of the parameter. Moreover, the parameter modifies the x-scale, therefore the speed of propagation. This interacting behavior is observed, for example, in real water waves, where higher-amplitude waves propagate at greater speeds. Top

Something very significant has simply and naturally fallen out of the commutative hypercomplex mathematics and the canonical formulation of our result. We have a complementary pair of periodic, traveling-wave solutions moving in opposite directions (of course, either of the pair could be blocked by the initiating boundary.) Apparently, because of the way the KdV equation is written and in the case of the 1-D eigenfunctions, above (whose negatives are not solutions of the KdV equation), we can have only positive-going spikes, here. However, remember that the most-general, 4-D solution u (Z), above, had a  ± sign in front of it, which for simple convenience we did not carry forward. Therefore, negative-going spikes are possible. Top

Actually, the sn(x,k) function is doubly periodic, and so our solution can model a field of wave spikes above a plane, such as the spiking wave swells seen under certain storm conditions on the ocean. Similarly for the 2-D array of quantum potential wells seen on the surface of a crystal lattice. Top

The infinite amplitude is clearly too idealized for real water waves. We could temper our solutions by doing orthogonal-function approximations of the sn^-2(x,k) terms with the same arguments, for example, ₁(x-ct), in such a way that the same general shape is maintained, but with finite amplitude and only near-discontinuous transition. Indeed, we could match the shape to what is observed in real wave experiments for specific fluids. With arguments as indicated, the approximation would then realistically exhibit the connection between overall wave amplitude and wave speed. We could then use the tailored solutions in numerical calculations for specific fluids. One such approximation is as follows. We note that sn²= 1-cn² and that 0 ≤ cn²(x,k) ≤ 1; therefore, we can use the rule for a simple geometric progression: Top

In these terms, our 1-D, sn^-2(x,k)-form solutions are converted to: Top

So far, we have made no approximations, but only an expansion. Note that the 1-D, sn^-2(x,k)-form eigenfunctions have periodic singularities, but no single term in the cn^2(n-1)(x,k)-form eigenfunctions exhibits any singularity. It is the sum of the terms that produces periodic singularities, as follows. Each cn^2(n-1)(x,k) term has amplitude unity and is superimposed and centered upon all of the previous terms. As one adds terms, one adds increments of amplitude by straight summation. Therefore, if we truncate the series, we are left with a finite-amplitude approximation of the original 1-D eigenfunctions. The truncated form would take on the appearance of an offset, pinched cn(x,k) function, thusly: Top

This is beginning to look like real water waves, and we can control the amplitude by the number of terms that we include in the truncation. Note, also, that throughout any truncations, we maintain the interaction between wave height and wave speed, such that higher-amplitude waves are propagated at higher speeds.

Next, we must account for the fact that the KdV equation also models solitons (solitary waves). To do so, we can set the arbitrary parameter k to unity and use the identity sn(x,1)=tanh(x) to convert our 1-D eigenfunctions to:

The tanh(x) function is an s-shaped curve with amplitude unity and only one zero crossing, at x=0. Its square has amplitude unity except for one negative-going spike that drops to zero, shown below: Top

As a result, the tanh^-2(x,t) eigenfunctions represent a complementary pair of positive-going, solitary wave spikes moving in opposite directions. Because of the above, each solitary spike has the following shape: Top

The offset base for the real tanh^-2(x,t) terms in the eigenfunctions is set by the square of the parameter. The amplitude is infinite, again pointing out the limitations of the KdV model for ordinary wave action. However, an infinite-amplitude solitary spike can be interpreted as a shock wave moving through the medium. An interesting experiment would be to incrementally manipulate the value of the k parameter in such a way as to show the transition between wave train action and solitary waves. Top

Similarly to what we did with the sn^-2(x,k)-form eigenfunctions, above, we can do an expansion and truncation of the tanh^-2(x,t) forms. We use the identity tanh²(x)=1-sech²(x) and the fact that 0 ≤ sech²(x) ≤ 1 to assert:

In these terms, our 1-D, tanh-form solutions are converted to: Top

Exactly as above, the tanh^-2(x,t) forms exhibit singularities, but no single term in the sech(x,t) expansion does so. As above, each sech^2(n-1)(x,t) term is superimposed and centered on all of the preceding ones, and each adds an increment of amplitude, eventually building to infinity. If we truncate the expansion after the sech⁴(x,t) term, for example, we can obtain a finite amplitude and a very realistic wave form that is often seen on the ocean: Top

Lastly, many published KdV solutions for particular problem conditions are based around the   sech²(x) function. This function is especially apropos for shallow-water, solitary wave behavior because it has a solitary hump shape and range 0 ≤ sech²(x) ≤ 1. Here, if we truncate after the first term, we are left with: Top

These forms have the same qualitative appearance as the sech⁴(x,t) truncation, above, but with a lower amplitude:

The relative height and width are adjustable via the parameter. For example: Top

These forms can represent low-hump waves (solitons) moving in opposing directions. The approximation is good because, although all the terms in the expansion have amplitude unity, we are raising 0 ≤ sech²(x) ≤ 1 fractional numbers to succeedingly higher powers, forcing them closer to zero. Therefore, each higher-order term represents a centered, superimposed, increasingly-narrower peak with amplitude unity. In a real water wave, their effect would be to increase the height of the wave and narrow its peak, trying to drive the amplitude to infinity. They even hint at the relative fluid motion within a wave as it is being formed. The shape approximation is best in the leading and trailing edges of the peak, because the higher-order terms have much narrower skirts than the sech²(x,t) term. It may well be that the higher-order terms are damped out in water, by processes and effects that are amenable to experiment. Top

We emphasize that all of the above idealized, propagating disturbances can occur only under well-prescribed circumstances. Basically, a single impulse or equally-spaced wavetrain impulses must propagate into a uniform, stationary medium. There must be no interferences, variable resistances, reflections, etc. If, on the other hand, there are multiple, random impulses at different points in the fluid, for example, then we must use a numerical, finite-element method of solution, using the characteristic functions given above in place of the original PDE. Top

As shown above, we can control wave height by truncating the series expansion. Further, we can even control the shape of the truncated form, as follows. We can run wave generation experiments measuring height, shape, and speed, then fit an approximating function to the data of the form:

In this approximation, the γ_i coefficients represent both a shaping and damping effect. For damping, they might decrease in magnitude as some function of the power of the succeeding sech(x,t) terms. For shaping, they might increase or decrease. They could be determined via a least-squares fit to experimental data. Once these are determined, one should then have a very accurate, specific model of the dynamics of a given fluid. These effects will vary from fluid to fluid, due to differing viscosity and other fluid properties. This type of experimental approximation can also be carried out on the sn^-2(x,k) form of the eigenfunctions. Top

Conclusions

We have found the most general characteristic function for the KdV equation when analyticity is imposed. We have integrated three times and have obtained three arbitrary constants of integration. In one general function, the result implicitly models traveling wave trains, shock waves, and solitons, all depending upon the selection of values for the three arbitrary constants. In all cases, the solution reduces to a pair of complementary solutions with wave actions moving in opposing directions. Either of the pair might be suppressed by the boundary conditions. The infinite amplitude of the solutions shows that the KdV equation overstates the possible and allowable real water wave height, but we have suggested a way to incorporate wave damping by fitting the characteristic function to real data.

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© 2003, 2008
Clyde M. Davenport
cmdaven@comcast.net