By C. Rogers and W.F. Shadwick (Eds.)
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Extra resources for Bäcklund Transformations and Their Applications
Example text
11 ) x'! 14) dx'" = 0, be integrable. 15) alEE'l ux -;I2-~=O. 17) where U, R, S, T, and V, in general, involve the quantities x", u, U;, and u' but not partial derivatives in u of second or higher order. 17), the well-known Monge-Ampere form is obtained. t The modern notion of a Backlund transformation in terms of a jet-bundle formalism is developed in Chapter 2. 16 1. 17). In particular, if the equations for u and u' so derived are both of the Monge-Ampere form, then the Backlund transformation may be regarded as a mapping between their integral surfaces.
14) dx'" = 0, be integrable. 15) alEE'l ux -;I2-~=O. 17) where U, R, S, T, and V, in general, involve the quantities x", u, U;, and u' but not partial derivatives in u of second or higher order. 17), the well-known Monge-Ampere form is obtained. t The modern notion of a Backlund transformation in terms of a jet-bundle formalism is developed in Chapter 2. 16 1. 17). In particular, if the equations for u and u' so derived are both of the Monge-Ampere form, then the Backlund transformation may be regarded as a mapping between their integral surfaces.
BACKLUND TRANSFORMATIONS AND NONLINEAR EQUATIONS is obtained. 132) shows that 'l,72 cP = o. 141) yields cPx h: + cP yh: + cPz = 0 at z = - h*(x, y). 151) augmented by the appropriate initial conditions. Attention is now restricted to the case of two-dimensional gravity waves propagating on the free surface of a layer of water bounded below by a horizontal, plane, impervious, and rigid base. It is convenient to introduce the vertical translation z --+ z + h* so that the lower boundary becomes z = O.