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6.7 The Korteweg–de Vries Equation  269
                                 ∂
                                   (A )= −A  im A mj .              (6.7.26)
                                     ij
          Let                  ∂e m

                                    ψ p =   A .                     (6.7.27)
                                              sp
                                          s
          Then, (6.7.26) can be written
                                         1
                            b r e r A A rj  = (b i + b j )A − ψ i ψ j .  (6.7.28)
                                                   ij
                                 ir
                                         2
                          r
          From (6.7.27) and (6.7.26),

                                     = −A  pq   A sq
                                 ∂ψ p
                                 ∂e q
                                              s
                                     = −ψ q A .                     (6.7.29)
                                             pq
          Let
                                            2
                                      θ p = ψ .                     (6.7.30)
                                            p
          Then,
                                      = −2ψ p ψ q A pq              (6.7.31)
                                  ∂θ p
                                  ∂e q
                                      =  ∂θ q  ,                    (6.7.32)
                                        ∂e p
                    2         ∂
                        = −2  ∂e p  (ψ q ψ r A )
                   ∂ θ r
                                       qr
                 ∂e p ∂e q
                        =2(ψ p ψ q A A qr  + ψ q ψ r A A rp  + ψ r ψ p A A ),
                                                                pq
                                                            rq
                                  pr
                                               qp
          which is invariant under a permutation of p, q, and r. Hence, if G pqr is any
          function with the same property,
                                  2

                                      =6     G pqr ψ p ψ q A A .    (6.7.33)
                                ∂ θ r
                                                       pr
                                                          qr
                           G pqr
                       p,q,r   ∂e p ∂e q  p,q,r
          The above relations facilitate the evaluation of the derivatives of v which,
          from (6.7.7) and (6.7.27) can be written

                                  v =   (ψ m − b m ).
                                      m
          Referring to (6.7.29),
                                  ∂v
                                               A mr
                                     = −ψ r
                                  ∂e r
                                            m
                                     = −ψ  2
                                          r
                                     = −θ r .                       (6.7.34)
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