1.5. When are two equations equivalent? 9 water waves on a canal and has since found application in many dif- ferent areas of science and engineering. Later, we will encounter the KP Equation (9.1), which is a theoretically important generalization of the KdV Equation. It will be useful for you to get to know these equations by name and to know properties or formulas for some of their solutions, much as one benefits from associating the equation a2 + b2 = c2 with the name “Pythagoras” and knowing that a = 3, b = 4, c = 5 gives one solution. However, especially if you are likely to consult other books or Internet resources, you may be disturbed to find that the differential equations bearing these names do not always look exactly look the same from source to source. We therefore need to broaden our view a bit to allow for the different choices of notation used by different authors. 1.5 When are two equations equivalent? It cannot be said that there is any universal rule that all mathe- maticians will recognize for saying when two differential equations are “equivalent”. However, for the purposes of this book4, we will say that two differential equations are just different ways to write the same equation if one can be turned into another by some combination of algebraic manipulation and variable changes of the form x ax+b where a and b are constants. Definition 1.4 We will say that a differential equation for the un- known function f(x1,...,xn) of the variables x1 through xn is equiv- alent to another equation for the function ˆ(X 1 , . . . , Xn) if one can find constants ai, bi for 0 i n with ai = 0 such that substituting f(x1,...,xn) = a0 ˆ(a 1 x1 + b1,...,anxn + bn) + b0 into the first equation (using the chain rule appropriately to replace derivatives of f with derivatives of ˆ), moving expressions from one side of the equation to the other, and multiplying the entire expression by some nonzero constant reproduces the second equation. 4 Keep in mind that there are more general equivalences which are used in other situations. For instance, (8.3) and (8.9) are both versions of the Sine-Gordon Equation, but are not considered equivalent according to Definition 1.4. Moreover, some even less restrictive notions of equivalence identify the KP Equation (9.1) and the Bilinear KP Equation (9.5).
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