The General Analytical Solution for the Korteweg-
de Vries Equation

© 2003, 2008

Clyde M. Davenport

cmdaven@comcast.net

Updated 8/2/08

Introduction | Hypercomplex Solution | Elementary Properties | Conclusions |

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 We shall present something new in the form of the
We emphasize 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
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
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 Making the partial derivative conversions, we obtain: where where The element From this, we see that an unavoidable condition
-( 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 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 This is the 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 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
The above solution is in terms of one 4-D independent variable,
The eigenvalues are classical complex variables, and the above definition means that a differential operator that operates on the 4-D variable i.e., the eigenfunctions of When we are given a 4-D expression such as
All of the subscripted components on the right (excluding the 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 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
This time, the individual vector components of
Elementary Properties We have obtained a solution 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=1 and
^{2}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.
TopOur reduced, one-dimensional solution 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 Consequently, the 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 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 Actually, the The infinite amplitude is clearly too idealized for real water waves. We could temper our solutions by doing orthogonal-function approximations of the _{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 sn
1-^{2}=cn and that
^{2}0 ≤ cn 1; therefore, we can use the rule for a simple geometric progression:
Top^{2}(x,k) ≤In these terms, our 1-D,
So far, we have made no approximations, but only an
expansion. Note that the 1-D, cn-form eigenfunctions exhibits any
singularity. It is the sum of the terms that produces periodic singularities, as
follows. Each ^{2(n-1)}(x,k)cn 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 ^{2(n-1)}(x,k)cn(x,k) function, thusly:
TopThis 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
The As a result, the The offset base for the real
k parameter in such a way as to show the transition between wave train
action and solitary waves.
TopSimilarly to what we did with the
tanh
forms. We use the identity
^{-2}(x,t)tanh1-^{2}(x)=sech
and the fact that ^{2}(x)0 ≤ sech1 to
assert:
^{2}(x) ≤ In these terms, our 1-D, Exactly as above, the sech(x,t) expansion does so. As above, each
sech 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
^{2(n-1)}(x,t)sech term, for example, we can obtain a
finite amplitude and a very realistic wave form that is often seen on the ocean:
Top^{4}(x,t)Lastly, many published KdV solutions for particular problem conditions are based around the 0 ≤ sech1. Here, if we truncate after
the first term, we are left with:
Top^{2}(x) ≤ These forms have the same qualitative appearance as the
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 sech 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^{2}(x,t)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 In this approximation, the 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 form of the eigenfunctions.
Top^{-2}(x,k)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.
© 2003, 2008 |