The myth about nonlinear differential equations

Taking the example of Koretweg–de Vries equation, it is shown that soliton solutions need not always be the consequence of the trade-off between the nonlinear terms and the dispersive term in the nonlinear differential equation. Even the ordinary one dimensional linear partial differential equation can produce a soliton. Solitary waves and solitons are often described [1-8] as a consequence of the tradeoff between nonlinear and dispersive terms in the nonlinear differential equations. This has given rise to the myth that solitary waves and solitons can be obtained as solutions of nonlinear differential equations only and not as solutions of linear differential equations. An associated misunderstanding is that only nonlinear differential equations are capable of describing nonlinear physical phenomena and nonlinear differential equations are more powerful than linear differential equations in describing physical phenomena. The observation that, in nature, linear phenomena are often only approximations to nonlinear phenomena probably gave birth to this belief. As a consequence, physicists are constructing more and more nonlinear differential equations to describe nonlinear physical phenomena. It is, of course true that nonlinear phenomena are more general, than linear phenomena. But linear differential equations are, in general neither approximations nor particular cases of nonlinear differential equations. Also, it is to be emphasised that solutions of both linear and nonlinear differential equations are functions which depend nonlinearly on the independent variable (the only exception being the straight line solution). We can construct linear as well as nonlinear differential equations from the same function. For example, from the function