Collected Papers on Wave Mechanics (Second Edition) by Erwin Schrödinger

By Erwin Schrödinger

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Extra info for Collected Papers on Wave Mechanics (Second Edition)

Example text

93) appears in [48] and the resulting nonlocal symmetries for the NLT equation U{x, t; u} (85) are summarized by the following theorem. Theorem 14 A point symmetry of the potential system UV1 {x, t; u, v1 } (86) yields a nonlocal symmetry of the NLT equation U{x, t; u} (85) if and only if the pair of constitutive functions (F(u), G(u)) satisfies the first order ODE system (c3 u + c4 )F (u) − 2(c1 − c2 − G(u))F(u) = 0 (c3 u + c4 )G (u) + G 2 (u) − (c1 − 2c2 + c3 )G(u) − c5 = 0 (94) in terms of arbitrary constants c1 , .

Correspondingly, one has the vector potential v = (v 0 , v 1 , v 2 ) and the underdetermined potential system given by Some Recent Developments in Finding 55 u t = vx2 − v 1y −u x = v 0y − vt2 (135) −u y = vt1 − vx0 . Now consider the equivalent augmented constrained system obtained by appending the Lorentz gauge (136) vt0 − vx1 − v 2y = 0 to the underdetermined potential system (135) to obtain the determined potential system u t = vx2 − v 1y −u x = v 0y − vt2 −u y = vt1 − vx0 (137) 0 = vt0 − vx1 − v 2y .

Yang Example 2 Nonlinear Wave Equation Now we use the nonlinear wave equation u tt − (c2 (u)u x )x = 0 (53) as an example to show how the Direct Method finds the fluxes for a local CL from a known local CL multiplier. In particular, one can show that Λ[U ] = xt is a local CL multiplier for the PDE (53). Then xt (Utt − (c2 (U )Ux )x ) = Dt (T [U ]) + Dx (X [U ]) (54) for some functions T [U ] = T (x, t, U, Ux , Ut ), X [U ] = X (x, t, U, Ux , Ut ). Consequently, the Eq. (54) becomes xt (Utt − 2c(U )c (U )Ux2 − c2 (U )Ux x ) = Tt + TU Ut + TUt Utt + TUx Ut x + X x + X U U x + X Ut U t x + X U x U x x .

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