9. (a) Use the Bcklund transformation (11.9.2ab) to show that four solutions u, (r=1,2,3,4) satisfy the...

  

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9. (a) Use the Bäcklund transformation (11.9.2ab) to show that four solutions u, (r=1,2,3,4) satisfy the relation tan - (- (- - -)} = (2+2) - { (-4,-2) - { tan- a₁-a₂ where a₁ and 92 are nonzero constants. (b) Apply the relation in Exercise 9(a) to obtain a soliton solution of the sine-Gordon equation by setting u, 0, ₂ and u, different soliton solutions. 10. Examine the asymptotic nature of the two-soliton solution (11.10.19), as →∞. 11. Derive the solitary wave solution of the sinh-Gordon equation = sinh“. (1101) of the - (u - v), 2 1 1 1 = a sin u + b), u+ 0), = = sin - (u - ), (11.9.2ab) 2 لرحلية هذه إمارس 2 م ما الله -2 9. (a) Use the Bäcklund transformation (11.9.2ab) to show that four solutions u, (r=1,2,3,4) satisfy the relation tan - (- (- - -)} = (2+2) - { (-4,-2) - { tan- a₁-a₂ where a₁ and 92 are nonzero constants. (b) Apply the relation in Exercise 9(a) to obtain a soliton solution of the sine-Gordon equation by setting u, 0, ₂ and u, different soliton solutions. 10. Examine the asymptotic nature of the two-soliton solution (11.10.19), as →∞. 11. Derive the solitary wave solution of the sinh-Gordon equation = sinh“. (1101) of the - (u - v), 2 1 1 1 = a sin u + b), u+ 0), = = sin - (u - ), (11.9.2ab) 2 لرحلية هذه إمارس 2 م ما الله -2