Ames, W.F

Jones, S.E

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 17, 484-487 (1967) 
Nonlinear Superposition* 
S. E. JONES AND W. F. AMES 
Department of Statistics and Computer Science, 
University of Delaware, Newark, Delaware 
Submitted by Richard Bellman 
1. INTRODUCTION 
The purpose of this paper is to introduce a concept of nonlinear super- 
position. We define the term “connecting function” and give several exam- 
ples. Many significant successes in constructing effective theories for physical 
phenomena can be traced to the linear principle of superposition. The ideas 
contained herein induce a similar tool for some nonlinear equations. 
2. CONNECTING FUNCTIONS 
For simplicity we restrict our attention to cases for two independent varia- 
bles. However, our remarks will apply to cases with more independent 
variables or to ordinary differential equations. 
DEFINITION. Let ui = ui(x, y), us = +(x, y) ,..., uk = u~(x, y) be solutions 
of some partial differential equation L(u) = 0. Then a function 
F = F(u, , u2 ,..., uK , X, y) is called a connecting function for L(U) = 0 if F 
is also a solution. This constitutes a nonlinear superposition principle. 
It is clear that a connecting function for a given equation is not necessarily 
unique. Consider the equation 
II, + u, = u. (2.1) 
This equation has the obvious connecting function 
F = u1 + us (2.2) 
* This research has been sponsored by the University of Delaware Research Founda- 
tion, Inc., and by the National Science Foundation under Grant GK-136. 
484 
NONLINEAR SUPERPOSITION 485 
since it is linear; however, 
F = [uln + uZn]lln (2.3) 
is also a connecting function for any real number n # 0. 
3. BASIC RESULTS 
Connecting functions for a class of quasilinear equations may be developed 
from a linear equation by dependent variable transformation. Consider the 
equation 
+, r) u, + &, r) u, = c(x, Y) 24. 
If we set u = F(er(x, y)), then (3.1) becomes 
(3.1) 
(3.2) 
Since (3.1) is linear, if u1 , ua ,..., uk: are solutions, then 
is also a solution. However, if I’ = F-l( U) and ui = F(vi), ua = F(v&.., 
uk = F(Q), we have 
V = F-‘[F(q) + F(Q) + *-* + F(v,t)l (3.3) 
is a solution to (3.2). We naturally assume that F-l exists. The basic equation 
(3.1) need not be first order or even linear. We could reason from a non- 
linear equation for which a connecting function has been found. 
Some very interesting combinations can be taken from (3.2) and (3.3). 
For example, if a = 1, b 3 1, c = 12 # 0, and F = vn, then (3.2) becomes 
(2.1) and (3.3) g ives us the connecting function (2.3). Another combination 
is c EE 1 and F = eV, which gives 
4% Y) % + @, Y) v, = 1 
and the connecting function obtained from (3.3) is 
F = log [exp (4 + exp (dl. 
(3.4) 
(3.5) 
A more complicated equation is 
Q(X, Y) % + b(x, Y) vy = 4% Y) wn (3.6) 
486 JONES AND AMES 
where n # 1. For this equation 
F = [log [exp (A vi+) + exp (-+& i+n)]ln/l”l--n’ (3.7) 
is a connecting function. Some of the results of this paragraph apply in 
more generality. 
4. BURGERS' EQUATION 
The equation 
ut + 1121, = Au,, (4.1) 
where h is a parameter, has been utilized by Burgers [2] as a mathematical 
model of turbulence. We give in this section connecting functions for this 
equation and an associated equation. 
By setting II = v, , integrating once with respect to x, and discarding an 
arbitrary function oft, (4.1) becomes 
vt+~v22=hvme, (4.2) 
an equation which has some significance in the burning of a gas in a rocket, 
see Forsythe and Wasow [3; p. 1411. If we now set v = - U log (w), (4.2) 
transforms to 
Wt = xw, (4.3) 
the one-dimensional diffusion equation. These transformations are already 
well-known. A complete discussion of them with proper priorities may be 
found in Ames [l ; p. 241. 
Reasoning with (4.3) as we did in Section 3, a connecting function for (4.2) 
is found to be 
F = - u log [exp (- -$) + exp (- -$)I . (4.4) 
Now, (4.4) can be used to find a connecting function for (4.1). This is seen to 
be 
F=-%&llog[exp(-&Jt UlK 4 dr) + exp (- & S:, 45, t) G)] 1 
(4.5) 
where x,, < x is some lower limit. 
NONLINEAR SUPERPOSITION 487 
5. REMARKS 
The method of Section 3 has been applied to equations other than (3.1) 
with success. In a private discussion, Professor J. R. Ferron has noted that 
as n -+ 0 in (2.3) the connecting function F = (u~u.#/~ is obtained. 
REFERENCES 
1. W. F. AMES. “Nonlinear Partial Differential Equations in Engineering.” Academic 
Press, New York, 1965. 
2. J. M. BURGERS. A mathematical model illustrating the theory of turbulence. 
Advan. Appl. Mech. 1 (1948), 171. 
3. G. E. FORWTHE AND W. R. WASOW. “Finite Difference Methods for Partial 
Differential Equations.” Wiley, New York, 1960. 


Nonlinear superposition 

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Nonlinear superposition

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