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Appendix B The Dirichlet-to-Neumann operator Due to the importance of the Dirichlet-to-Neumann operator (DtN 0 ) in modelling water waves we pointed here its relation with the Hilbert transform on the flat strip used througout this work. Given the problem ⎧ ⎪⎨ ⎪⎩ z υυ + z ωω = 0, −h≤ω≤0, z ω = g(υ), ω=0, z ω = 0, ω=−h, the operatorT returns the tangential derivative z υ (υ, 0) from the Neumann data z ω (υ, 0)=g(υ), that is T [g]=z υ (υ, 0). Therefore, its inverseT satisfies g(υ)=z ω (υ, 0)=T [z υ (υ, 0)]. (B.1) 91
The inverseT ofT has a symbol−i tanh (kh). Alternatively it can be defined by the principal value integral T [ f ]=− 1 2h f (v) sinh ( π(v−υ)/2h ) dv. (B.2) The DtN 0 for the problem ⎧ ⎪⎨ ⎪⎩ z υυ + z ωω = 0, −h≤ω≤0, z(υ, 0)= f (υ), z(υ,−h)=0, returns the normal derivative (Neumann condition) from the Dirichlet data. In light of (B.1) we have DtN 0 [ f ]=T [ f ′ ]. It means that the Dirichlet-to-Neumann operator applies one differentiation plus the inverse operatorT to the Dirichlet data in order to obtain the Neumann condition. 92
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INSTITUTO NACIONAL DE MATEMÁTICA P
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Acknowledgements First, I would lik
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4.1 Hierarchy of one-dimensional mo
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Resumo É obtido um modelo reduzido
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when the steepening of a given wave
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in particular that of solitary wave
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Chapter 2 Derivation of the reduced
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demands that η t + u i η x = w i
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function f (x, z, t), let its assoc
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Substitution of w 1 into Eq. (2.2)
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i. e. u 1z =βw 1x . Hence ( ) u (0
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fluid layer. 2.2 Connecting the upp
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Furthermore, [√ ( √ )] βφt x,
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The problem in conformal coordinate
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problem (2.22). Therefore, P x =−
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assumption was made on the wave amp
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Its linearization around the undist
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c 0 chosen, there will be a right-
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and for similar reasons η t +η x
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3. For system (2.28) a similar unid
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Chapter 3 A higher-order reduced mo
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expression above as Let us express
Page 45 and 46: and substituting in the asymptotic
Page 47 and 48: and it is easy to take x-derivative
Page 49 and 50: which together with η tt = ( (1−
Page 51 and 52: For the weakly nonlinear model of o
Page 53 and 54: Again, ω2 k 2 large. → 0 as k→
Page 55 and 56: 2.6 2.4 2.2 ω/k 2 1.8 1.6 1.4 1.2
Page 57 and 58: Chapter 4 Numerical results 4.1 Hie
Page 59 and 60: Linear Weakly nonlinear Strongly no
Page 61 and 62: is to approximate theξ-derivative
Page 63 and 64: inary axis (where the eigenvalues o
Page 65 and 66: 2.4 2.2 RK4 AM3 AM4 2 1.8 1.6 |R(z)
Page 67 and 68: Also, recall that K1=E(η n , u n 1
Page 69 and 70: • T is the convolution matrix for
Page 71 and 72: The AM4 scheme used is: η n+1 j V
Page 73 and 74: 0.6 0.4 0.2 η 0 −0.2 40 35 30 25
Page 75 and 76: 0.6 0.5 0.4 0.3 η 0.2 0.1 0 −0.1
Page 77 and 78: where c 1 and c 2 are two functions
Page 79 and 80: 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 81 and 82: 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 83 and 84: 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 85 and 86: 0.6 0.5 0.4 0.3 η 0.2 0.1 0 −0.1
Page 87 and 88: 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0
Page 89 and 90: −η 0.1 0.08 0.06 0.04 0.02 0 60
Page 91 and 92: Conclusions and future work In the
Page 93 and 94: Appendix A Approximation for the ho
Page 95: thenω 0 satisfies the Neumann prob
Page 99 and 100: Setξ=πξ/l. In the new coordinate
Page 101 and 102: Takingξ-derivatives, φ ξ (ξ,ζ)
Page 103 and 104: Bibliography [1] Artiles, W. & Nach
Page 105 and 106: [16] Koop, C. G. & Butler, G., 1981
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