Determinants and Their Applications in Mathematical Physics
266 6. Applications of Determinants in Mathematical Physics c. Dt(φij)=φi 0 φj 2 −φi 1 φj 1 +φi 2 φj 0 − 1 2 (φi+3,j+φi,j+3). Pr ...
6.7 The Korteweg–de Vries Equation 267 =0+ 1 2 ∑ p,r (b i+2 p b j r −b j+2 p b i r )A pr = 1 2 (φi+2,j−φi,j+2), which is identic ...
268 6. Applications of Determinants in Mathematical Physics 6.7.3 The First Form of Solution, Second Proof Second Proof of Theor ...
6.7 The Korteweg–de Vries Equation 269 ∂ ∂em (A ij )=−A im A mj . (6.7.26) Let ψp= ∑ s A sp . (6.7.27) Then, (6.7.26) can be wri ...
270 6. Applications of Determinants in Mathematical Physics Hence, vx=− ∑ r brer ∂v ∂er = ∑ r brerθr. (6.7.35) Similarly, vt=− ∑ ...
6.7 The Korteweg–de Vries Equation 271 =− 2 ∑ p,q b 2 pbqepeqψpψqA pq , S=2R. (6.7.40) Referring to (6.7.33), (6.7.28), and (6.7 ...
272 6. Applications of Determinants in Mathematical Physics Proof. D j x (φi)= ( 1 2 bi )j e − 1 / 2 i [(−1) j λiei+μi] so that ...
6.7 The Korteweg–de Vries Equation 273 Hence, from (6.7.48), K (n) ij Un = μj λi [ H (n) ij Xn ] yi=−xi=bi = μj λi ∏n p=1 (bp+bj ...
274 6. Applications of Determinants in Mathematical Physics into the KdV equation yields ut+6uux+uxxx=2Dx ( F w 2 ) , (6.7.56) w ...
6.7 The Korteweg–de Vries Equation 275 Hence,wwzz−w 2 z =0, F=wwxt−wxwt+3w 2 xx − 4 wxwxxx+wwxxxx+3(wwzz−w 2 z ) =w [ (wt+4wxxx) ...
276 6. Applications of Determinants in Mathematical Physics wxxxx=2Vn− 3 ,n,n+1+3Vn− 3 ,n− 1 ,n+2+3Vn− 2 ,n− 1 ,n+1+Vn− 3 ,n− 2 ...
6.8 The Kadomtsev–Petviashvili Equation 277 − ∑ r αrA (n) r,n− 1 ∑ s βsA (n) sn = ∑ r ∑ s αrβs [ AnA (n) rs;n− 1 ,n− ∣ ∣ ∣ ∣ ∣ A ...
278 6. Applications of Determinants in Mathematical Physics Applying (A), v=Dx(logA)=− ∑ r λrerA rr , (6.8.3) Dy(logA)= ∑ r λrμr ...
6.8 The Kadomtsev–Petviashvili Equation 279 new formulae for the derivatives of logA: v=Dx(logA)= ∑ r,s A rs − ∑ r λr, (6.8.15) ...
280 6. Applications of Determinants in Mathematical Physics −Hi+2,j+Hi,j+2. (6.8.20) From (6.8.15), v=h 00 −constant. The deriva ...
6.9 The Benjamin–Ono Equation 281 Theorem.The KP equation in the form (6.8.2) is satisfied by the Wronskianwdefined as follows: ...
282 6. Applications of Determinants in Mathematical Physics 6.9.2 Three Determinants.................. The determinantAand its c ...
6.9 The Benjamin–Ono Equation 283 two columns as follows: P= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ c 1 1 c 2 1 ··· ··· [aij]n ··· ··· ·· ...
284 6. Applications of Determinants in Mathematical Physics = ∑ r ∑ s crcsArs. (6.9.13) The determinantsA,B,P, andQ, their cofac ...
6.9 The Benjamin–Ono Equation 285 To prove (6.9.17), perform the row operations (6.9.24) on Pn+1,n+2 and apply (6.9.7). To prove ...
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