# q-Advanced Models for Tsunami and Rogue Waves

Authors: D. W. Pravica; N. Randriampiry; M. J. Spurr
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A waveletKq(t), that satisfies the q-advanced differential equation Kq′(t)=Kq(qt) for q>1, is used to model N-wave oscillations observed in tsunamis. Although q-advanced ODEs may seem nonphysical, we present an application that model tsunamis, in particular the Japanese tsunami of March 11, 2011, by utilizing a one-dimensional wave equation that is forced by Fq(t,x)=Kq(t)qSin(x). The profile Fq is similar to tsunami models in present use. The function Sin(t)q is a wavelet that satisfies a q-advanced harmonic oscillator equation. It is also shown that another wavelet,  Cos(t)q, matches a rogue-wave profile. This is explained in terms of a resonance wherein two small amplitude forcing waves eventually lead to a large amplitude rogue. Since wavelets are used in the detection of tsunamis and rogues, the signal-analysis performance of Kq and Cosq is examined on actual data.


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## Body

## 1. Introduction

Tsunami or maremoto waves occur in response to earthquakes or landslides on the seafloor of large bodies of water, as discussed in [1–4]. The consequential runup to the shore is such that the tide goes out, then returns as a large surge, only to be followed by several diminishing cycles of similar events [5]. An understanding of this behavior involves a consideration of the effects from the seafloor near the shore where the wave velocity decreases [6, 7]. We study a short-lived forcing that predominantly generates a traveling-wave profile Kq(x-ct) where Kq solves a multiplicatively advanced differential equation (MADE) whose profile resembles a typical tsunami (see Figure 1). In contrast to the N-wave profile proposed in [8], the Kq-MADE profiles are asymmetric wavelets that are flat on a half-line.(a) Tsunami 52406 from DART, March 2011; (b) replica of tsunami-model profile adapted from [7]; (c) MADE solution profile y=Kq(t) for q=1.5.
(a)
(b)
(c)In an apparently unrelated phenomena, rogue, freak, or monster waves are caused by small ripples or currents in layers near the water's surface [9, 10]. Various methods have been used to model these rare events, as in [11, 12]. Here, we demonstrate why rogue waves may be a type of resonance wherein an arbitrarily low amplitude forcing, for a sufficiently long period of time, can produce any size of localized wave. Our models use square-integrable versions of the sine and cosine functions that we call Cos(t)q and Sin(t)q (see Figure 2).(a)y=Cos(t)q for q=1.5; and  (b) y=Sin(t)q for q=1.5.
(a)
(b)Tsunami and rogue waves are perturbations of the water-surface elevation functionη(t,x,y). The surface velocity components U≡(u,v,w) are small and the time-averaged mean-flow velocity 〈U〉 is assumed to remain 0 everywhere. Subject to assumptions, the functions Kq, Cosq and Sinq will be used in the construction of the forcing terms in the Matsuno equations [13] as given by
(1.1)∂u∂t+u∂u∂x+v∂u∂y-f0v=-α(u,v)u-g0(∂η∂x)+Px,∂v∂t+u∂v∂x+v∂v∂y+f0u=-α(u,v)v-g0(∂η∂y)+Py,w+∂∂x[(H+η)u]+∂∂y[(H+η)v]=-α(u,v)η+Q,
where H(x,y) is the depth of the water, w≡∂η/∂t is the vertical velocity of the wave surface, Px and Py are variable external forcings, and Q is a mass source term. The acceleration due to gravity g0 accounts for buoyancy, and the Coriolis parameter f0 addresses the rotation of the earth (only required for very long waves over the earth [14]). The friction coefficient α≥0 acts as a generic sink that reduces the amplitude over time [6].This work considers localized plane waves, or wavelets, on a flat sea propagating only in thex-direction. Thus, we set
(1.2)u=u(t,x),w=w(t,x),η=η(t,x),v(t,x,y)=0,H=H(x),c=c(x)≡g0H(x),Px=Px(t,x),α=f0=Py=Q=0.
In deep water H=H0>0 is a fixed constant and c0=g0H0 is a constant surface-wave speed (called the celerity). Then, for small amplitudes, the elevation function η(t,x) satisfies
(1.3)generation:∂2η∂t2-c02∂2η∂x2=F(t,x),whereη(t,x)≃0,fort≪0,x∈ℝ,(1.4)propagation:∂η∂t+c0∂∂x(η+βc0η2+νc0∂2η∂x2)=0,forH(x)≃H0,(1.5)runup:∂η∂t+c∂η∂x+η2dcdx+β∂(η2)∂x+ν∂3η∂x3=0,forH(x)>0.
These equations are found in [8, Equation (2)], [15, Equation (45)], and [16, Equation (13)], respectively. The new coefficients are defined as β(x)≡3c/(4H) and ν(x)≡cH2/6. Equation (1.4) is referred to as the KdV-top model in [16]. Inclusion of an additional boundary-turbulence term in (1.5) would result in a loss of conservation properties for η (considered in [17] but not here).The forcingsF(t,x) in (1.3) will be constructed from bounded wavelets ϕ(u) that satisfy MADEs. As such they satisfy the following conditions:
(1.6)ϕ∈ℒ1(ℝ)∩ℒ2(ℝ)∩ℒ∞(ℝ),∫-∞∞‍ϕ(u)du=0.
Physically, if ϕ represents the displacement of the water-level from equilibrium, conservation of mass necessitates that the conditions in (1.6) hold. Note that, as discussed in [15], solitons are not expected to occur in (1.4) since the total spatial integral of η vanishes. We find that if F(t*,x) is a wavelet in the x variable (for fixed t*∈ℝ), then so is η(t*,x) away from the shore (using x-integration of either (1.3) or (1.4)). During the runup, however, η(t,x) looses its wavelet properties due to the variability of the coefficients in (1.5).It seems natural that wavelets should appear in the study of surface water waves. The wavelets presented here are particularly well suited for surface waves. In particular, we show that when the forcingF is expressed in terms of Kq, Cosq and Sinq in (2.2)-(2.3), the solution can be reexpressed in terms of these functions. An objective of this paper is to demonstrate that the modeling as well as the detection and analysis of an observed wave profile can be achieved efficiently in terms of the wavelets (2.2)-(2.3), see [18–20]. Hence, a brief discussion on the topics of signal analysis and recovery on real data is presented using Kq and Cosq. We apply these techniques to the Japanese tsunami of March 11, 2011. The paper is completed with the details of a perturbation analysis in q>1+ that is needed to establish the existence of a resonance for rogue waves.


## 2. Preliminaries on Special Functions

The first quantity that we introduce is the functionKq(t) that satisfies the MADE
(2.1)Kq′(t)=Kq(qt),∀t∈ℝ,q>1.
We set Kq(t)≡0 for t≤0, and define
(2.2)Kq(t)≡∑j=-∞∞‍(-1)je-qjtqj(j+1)/2,fort>0,q>1.
Note that Kq(t) satisfies the wavelet conditions in (1.6), see [18]. Furthermore, as in [20], the reproducing kernel of Kq(t) gives the functions
(2.3)Cos(t)q=Nq·∫0∞‍Kq(u)Kq(u-t)du=Nq∑j=-∞∞‍(-1)je-q2j|t|qj2,Sin(t)q=-Nq·∫0∞‍Kq(u)Kq(qu-qt)du=(t|t|)Nq∑j=-∞∞‍(-1)je-q2j|t|qj(j-1),
which are displayed in Figure 2 for q=1.5. The normalization constant Nq is chosen so that Cos(0)q=1. It is shown in [20] that the q-advanced harmonic oscillator equations,
(2.4)Cos′′(t)q=-q·Cos(qt)q,Sin′′(t)q=-q2·Sin(qt)q,
hold, which are second-order MADEs. Each of (2.2)-(2.3) lies in ℒ2. We show as a consequence of Theorem 2.1 that Cos(t)q→cos(t) and Sin(t)q→Sin(t) pointwise for t∈ℝ as q→1+. Note, we also have uniform convergence on compact sets, which was obtained in [20]. Thus, Cos(t)q and Sin(t)q can be viewed as ℒ2 approximations of cos(t) and sin(t),  respectively, which are solutions of the limit of equations in (2.4) as q→1+.


### 2.1. Theta Functions

As part of our study, we employ the Jacobi theta function, defined forω∈ℂ∖{-qn}n∈ℤ,
(2.5)θ(q,ω)≡∑k=-∞∞ωkqk(k-1)/2=μq∏n=0∞(1+ωqn)(1+1ωqn+1),μq≡∏n=0∞(1-1qn+1).
This function will only be used in regions where ω>0 and q>1. Clearly the function θ(q2,ω2) is C∞ for all ω∈ℝ-{0} and grows faster than any rational polynomial at ±∞ and 0. As such, for each q>1 one has that 1/θ(q2,ω2) is a Schwartz function that is flat at ω=0. It also satisfies the algebraic identity
(2.6)θ(q;qnω)=qn(n+1)/2ωnθ(q;ω),n∈ℤ,
which plays an important role in our analysis, [18].Theorem 2.1.
Forq>1, let
(2.7)δq(ω)≡{1[ln(q)θ(q2;ω2)]ifω>00ifω≤0.
Then δq(ω) is a delta sequence in q>1+, at ω=1 and has the following properties:
(2.8)(i)limq→1+δq(ω≠1)=0,(ii)limq→1+δq(1)=∞,(iii)limq→1+∫0∞‍δq(ω)dω=1.Proof.
From [21], for 1<q<eπ and for ω>0 one has
(2.9)0<q1/4|ω|e(ln(|ω|))2/ln(q){πln(q)-1}≤θ(q2;ω2),(2.10)0<θ(q2;ω2)≤q1/4|ω|e(ln(|ω|))2/ln(q){πln(q)+1}.
Multiplication of (2.9) and (2.10) by ln(q), followed by reciprocation, gives that for ω>0(2.11)[q-1/4|ω|-1e-(ln(|ω|))2/ln(q)ln(q){π+ln(q)}]≤δq(ω)≤[q-1/4|ω|-1e-(ln(|ω|))2/ln(q)ln(q){π-ln(q)}].
Now, (i) follows from (2.11) and the fact that for k>0, limq→1+[e-k/ln(q)/ln(q)]=0. For ω=1, (ii) also follows from (2.11). To obtain (iii), we handle some preliminaries. First, it was shown in [18] that
(2.12)∥Kq∥22≡∫-∞∞‍|Kq(t)|2dt=∫-∞∞‍q2(μq)4μq22πθ(q2;q2ω2)dω=q(μq)4μq22π∫-∞∞‍dvθ(q2;v2)=q(μq)4μq2π∫0∞‍dvθ(q2;v2).
The functional identity (μq)4q=2∥Kq∥22(μq2)2Nq, established in [20], allows for the replacement of (∥Kq∥22/q) by ((μq)4/[2(μq2)2Nq]) in (2.12) to obtain the following:
(2.13)∫0∞‍dvθ(q2;v2)=π∥Kq∥22q(μq)4μq2=π2Nq(μq2)3.
This in turn gives that
(2.14)limq→1+∫0∞‍δq(ω)dω=limq→1+∫0∞‍dωln(q)θ(q2;ω2)=limq→1+π2ln(q)Nq(μq2)3=1,
where the last equality follows from the q-Wallis limit, limq→1+[ln(q)Nq(μq2)3]=π/2, from [21]. This proves (iii) and finishes the proof of the theorem.


### 2.2.q-Advanced Wavelets That Solve MADEs

The appearance of theta functions is a consequence of the Laplace transform of (2.2),
(2.15)Kq(t)=ℒ-1[-μq3sθ(q,s)](t),whereℒ[f(t)](s)≡∫t=0∞‍e-stf(t)dt.
In [20], we use the inverse Fourier transform to compute the expressions
(2.16)Cos(t)q=(μq2)3Nqπ∫-∞∞‍eiωtdωθ(q2,ω2),Sin(t)q=-i(μq2)3Nqπ∫-∞∞‍eiωtωdωθ(q2,ω2),
and these expressions will be used in the proof of a resonance for rogue waves. The representation in (2.15) verifies the MADE in (2.1), and (2.16) implies the identities
(2.17)Cos′(t)q=-Sin(t)q,Sin′(t)q=q·Cos(qt)q,
which verify the MADEs in (2.4). Now Theorem 2.1 gives that both δq(ω) and δq(-ω) are delta sequences in q>1+ at ω=±1, respectively. These applied to (2.16), in combination with the q-Wallis limit, give that pointwise
(2.18)limq→1+Cos(t)q=cos(t),limq→1+Sin(t)q=sin(t),fort∈ℝ.
For q>1, the functions in (2.15) and (2.16) are in the Schwartz space. In particular, their rates of decay for large |t| are typically slower than exponential but faster than any reciprocal polynomial. It can be shown that ∃C>0 so that for t∈ℝ and q>1,(2.19)max{|Kq(t)|,|Cos(t)q|,|Sin(t)q|,1|θ(q;t)|}≤Cq-logq2(|t|).Remark 2.2.
The use of theta functions in frequency space provides decaying versions ofcosine and sine while preserving many differential properties of these functions.


## 2.1. Theta Functions

As part of our study, we employ the Jacobi theta function, defined forω∈ℂ∖{-qn}n∈ℤ,
(2.5)θ(q,ω)≡∑k=-∞∞ωkqk(k-1)/2=μq∏n=0∞(1+ωqn)(1+1ωqn+1),μq≡∏n=0∞(1-1qn+1).
This function will only be used in regions where ω>0 and q>1. Clearly the function θ(q2,ω2) is C∞ for all ω∈ℝ-{0} and grows faster than any rational polynomial at ±∞ and 0. As such, for each q>1 one has that 1/θ(q2,ω2) is a Schwartz function that is flat at ω=0. It also satisfies the algebraic identity
(2.6)θ(q;qnω)=qn(n+1)/2ωnθ(q;ω),n∈ℤ,
which plays an important role in our analysis, [18].Theorem 2.1.
Forq>1, let
(2.7)δq(ω)≡{1[ln(q)θ(q2;ω2)]ifω>00ifω≤0.
Then δq(ω) is a delta sequence in q>1+, at ω=1 and has the following properties:
(2.8)(i)limq→1+δq(ω≠1)=0,(ii)limq→1+δq(1)=∞,(iii)limq→1+∫0∞‍δq(ω)dω=1.Proof.
From [21], for 1<q<eπ and for ω>0 one has
(2.9)0<q1/4|ω|e(ln(|ω|))2/ln(q){πln(q)-1}≤θ(q2;ω2),(2.10)0<θ(q2;ω2)≤q1/4|ω|e(ln(|ω|))2/ln(q){πln(q)+1}.
Multiplication of (2.9) and (2.10) by ln(q), followed by reciprocation, gives that for ω>0(2.11)[q-1/4|ω|-1e-(ln(|ω|))2/ln(q)ln(q){π+ln(q)}]≤δq(ω)≤[q-1/4|ω|-1e-(ln(|ω|))2/ln(q)ln(q){π-ln(q)}].
Now, (i) follows from (2.11) and the fact that for k>0, limq→1+[e-k/ln(q)/ln(q)]=0. For ω=1, (ii) also follows from (2.11). To obtain (iii), we handle some preliminaries. First, it was shown in [18] that
(2.12)∥Kq∥22≡∫-∞∞‍|Kq(t)|2dt=∫-∞∞‍q2(μq)4μq22πθ(q2;q2ω2)dω=q(μq)4μq22π∫-∞∞‍dvθ(q2;v2)=q(μq)4μq2π∫0∞‍dvθ(q2;v2).
The functional identity (μq)4q=2∥Kq∥22(μq2)2Nq, established in [20], allows for the replacement of (∥Kq∥22/q) by ((μq)4/[2(μq2)2Nq]) in (2.12) to obtain the following:
(2.13)∫0∞‍dvθ(q2;v2)=π∥Kq∥22q(μq)4μq2=π2Nq(μq2)3.
This in turn gives that
(2.14)limq→1+∫0∞‍δq(ω)dω=limq→1+∫0∞‍dωln(q)θ(q2;ω2)=limq→1+π2ln(q)Nq(μq2)3=1,
where the last equality follows from the q-Wallis limit, limq→1+[ln(q)Nq(μq2)3]=π/2, from [21]. This proves (iii) and finishes the proof of the theorem.


## 2.2.q-Advanced Wavelets That Solve MADEs

The appearance of theta functions is a consequence of the Laplace transform of (2.2),
(2.15)Kq(t)=ℒ-1[-μq3sθ(q,s)](t),whereℒ[f(t)](s)≡∫t=0∞‍e-stf(t)dt.
In [20], we use the inverse Fourier transform to compute the expressions
(2.16)Cos(t)q=(μq2)3Nqπ∫-∞∞‍eiωtdωθ(q2,ω2),Sin(t)q=-i(μq2)3Nqπ∫-∞∞‍eiωtωdωθ(q2,ω2),
and these expressions will be used in the proof of a resonance for rogue waves. The representation in (2.15) verifies the MADE in (2.1), and (2.16) implies the identities
(2.17)Cos′(t)q=-Sin(t)q,Sin′(t)q=q·Cos(qt)q,
which verify the MADEs in (2.4). Now Theorem 2.1 gives that both δq(ω) and δq(-ω) are delta sequences in q>1+ at ω=±1, respectively. These applied to (2.16), in combination with the q-Wallis limit, give that pointwise
(2.18)limq→1+Cos(t)q=cos(t),limq→1+Sin(t)q=sin(t),fort∈ℝ.
For q>1, the functions in (2.15) and (2.16) are in the Schwartz space. In particular, their rates of decay for large |t| are typically slower than exponential but faster than any reciprocal polynomial. It can be shown that ∃C>0 so that for t∈ℝ and q>1,(2.19)max{|Kq(t)|,|Cos(t)q|,|Sin(t)q|,1|θ(q;t)|}≤Cq-logq2(|t|).Remark 2.2.
The use of theta functions in frequency space provides decaying versions ofcosine and sine while preserving many differential properties of these functions.


## 3. Tsunami Modeling Using MADEs

A tsunami wave is the consequence of a spontaneous change in elevation on the seafloor, which creates a variable pressure field throughout the volume of water. This sets up forces that extend to the surface of the ocean causing it to be moved up and down, locally. The perturbed wave height then propagates away from this disturbance. In still water, the surface wave speedc0=g0H0 mainly depends on the average depth H0 of the ocean. For tsunamis, the wavelength is much longer than H0 and so the shallow-water wave equation  applies (see [22, page 195]). Near the shore nonlinear effects need to be introduced to account for the sloping of the shoreline [4, 12, 23, 24].Suppose that the faultline on the seafloor is parallel to they coordinate. Then the y-dependence can be ignored even after the tsunami-causing event has taken place. In this situation, the wave front will travel in the x direction only.


### 3.1.q-Advanced Tsunami Model

Consider the forced one-dimensional wave equation for the water-level functionψ [14, 22],
(3.1)∂t2ψ(t,x)-c02∂x2ψ(t,x)=F(t,x),∀t∈ℝ,x∈ℝ,ψ(t,x)=0,∀t<0,x∈ℝ,
where the condition on the right in (3.1) constitutes the boundary and initial conditions. The forcing is expected to satisfy the conditions, for fixed t*, x*∈ℝ,
(3.2)lim|t|→∞F(t,x*)=0,lim|x|→∞F(t*,x)=0,∫-∞∞‍F(t*,x)dx=0,
and is related to the depth of the ocean floor by ∂t2H(t,x)=F(t,x). The models in [4, 8] start with ground-motion profiles H=HTS or H=HZWL where
(3.3)HTS≡Aγtanh(γx-tτ),HZWL≡(2Aγ)sech2(γx)tanh(γx)sin(πt2τ),
respectively. In [8], it is noted that using the first model H=HTS gives a force consistent with a landslide that continues for all time. In [4], the second model H=HZWL suggests an earthquake that continues for all time. In these settings, (3.2) does not hold. However, these models are still used as part of the initial forcings for the wave equation since they lead to integrable solutions whose evolutions resemble that of actual tsunamis. As a comparison, we propose the following q-advanced model:
(3.4)Hq(t,x)≡A·Kq(tτ)Sin(γ·x)q,Fq(t,x)≡∂2Hq∂t2=A·qτ2Kq(q2tτ)Sin(γ·x)q,
where Fq(t,x) now satisfies the conditions in (3.2). When (3.4) is substituted into (3.1), solved and simplified, one obtains a unique solution ψ(t,x) to the forced wave equation. To express the solution, define the two phase functions, corresponding to right and left propagation,
(3.5)φμ+(t,x,ξ)≡tτ-γ·x-ξqμ,φμ-(t,x,ξ)≡tτ+γ·x-ξqμ,∀t,x,ξ∈ℝ,
where the parameters are related by c0γτ=qμ. Define, for any α,β,μ∈ℝ and q>1,
(3.6)𝒯α,β,μ±(t,x)≡∫ξ=0γ·x‍Sin(qα·ξ)q·Kq(qβ·φμ±(t,x,ξ))dξ.
Then, for t>0 and x∈ℝ, the reader can verify that
(3.7)ψ(t,x)=ψparticular(t,x)+ψhomogeneous(t,x),
where a particular solution to (3.1), with the forcing in (3.4), is
(3.8)ψparticular(t,x)=(-A·q-μ2)·[𝒯0,1,μ-(t,x)-𝒯0,1,μ+(t,x)].
A smooth solution that satisfies ψ(t,x)=0, for t≤0, is obtained by choosing
(3.9)ψhom.(t,x)=-12[ψpar.(0,x+c0t)+ψpar.(0,x-c0t)+∫x-c0tx+c0t‍∂ψpar.(0,u)c0·∂tdu].
One can see from (3.5)–(3.9) that the solution has the basic wavelet properties
(3.10)ψ(·,x*),ψ(t*,·)∈ℒ1(ℝ)∩ℒ2(ℝ)∩ℒ∞(ℝ),∫ℝ‍ψ(t,x*)dt=∫ℝ‍ψ(t*,x)dx=0,
for each t*,x*∈ℝ. To analyze the long-term behavior of the solution ψ, we use (2.1), (2.17), and integration by parts, twice, to obtain Proposition 3.1.Proposition 3.1.
For anyα,β,μ∈ℝ and q>1 the following identity holds:
(3.11)𝒯α,β,μ±(t,x)=q-α·Kq(qβ·[tτ∓γxqμ])-q-2α+2β-2μ+1·𝒯α-1,β+2,μ±(t,x)-q-α·Cos(qαγx)q·Kq(qβtτ)±q-2α+β-μSinq(qα-1γx)·Kq(qβ+1tτ).The expression in (3.11) demonstrates that 𝒯α,β,μ±, as used in (3.8), can be written as a series of localized (bound) states and traveling (free) states. These different terms appear as translated and scaled versions of the wavelets Kq, Cosq and Sinq. Thus, such functions provide a good match for a wave profile ψ that was generated by the family of forcings Fq in (3.4). Furthermore, for t≫0, x≫0 and μ≫0 (reasonable for tsunamis),
(3.12)ψparticular(t,x)≃(Aq-3μ-12)·Kq(γ·q-μ[c0t-x]).
Finally, observe that Fq(t,x) in (3.4) is q-advanced in time compared to the ground-motion profile Hq(t,x). We also find that the forcing precedes the response ψ for the q-advanced models. Figure 4(a) illustrates a case where a forcing profile generates a similarly shaped tsunami in Figure 4(b).Remark 3.2.
Similar results hold for height functionsH~q(t,x)≡A~·Kq(t/τ)Cos(γ·x)q.


### 3.2. Numerical Solution of aq-Advanced Tsunami Wave Event

Here, we model the Japan tsunami of March 11, 2011, using (1.3), (1.4), and (1.5) with a forcing term Fq in (3.4) with environmental parameters chosen to be, in mks units,
(3.13)g0=10ms-2,H0=4000m,c0=200ms-1,γ=0.01m-1,τ=100s,A=6×107m,
which are based on past experience [4]. The sea-depth function is chosen to be
(3.14)H(x)=H0,forx≤0,H(x)=H0·[1-(x5km)2],forx∈(0,5km),
which models the sea-floor near Wake Island. This is a required modification of the model used in [7]. For stability, we employ a Lax-Wendroff correction term in the numerical method. A good match with typical tsunami profiles and run-up profiles was obtained by using q=1.25. In this case, μ≃23.74, which justifies the approximation in (3.12).Figure3 represents data from the Japan tsunami of March 11, 2011, at the first oceanic observation site DART 21418. This data includes a preliminary forcing profile from the (local) times 5 min to 20 min, along with the actual tsunami profile from the (local) times 42 min to 72 min.Figure 3
DART 21418: Data showing earthquake (forcing, on the left) and resulting tsunami (on the right) observed on March 11, 2011, from 5:43 am to 6:51 am (horizontal axis is time in minutes, vertical axis is in meters).(a) Comparison offorcing from data (dashed line) and model (solid line) results in q=1.25; (b) comparing tsunami at DART 21418 and propagation model results in A=6×107m.
(a)
(b)Figure4(a) again shows the Dart 21418 forcing profile from the times 5 min to 20 min, along with a q-advanced forcing profile as in (3.4), with A and q chosen to be A=6×107 and q=1.25 to effect comparable forcing profiles.Figure4(b) shows the Dart 21418 actual tsunami profile from the times 42 min to 72 min along with the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing Fq in (3.4) as above (with parameters A=6×107 and q=1.25).Figure5 shows the actual tsunami profile at a later time and greater distance at oceanic observation DART 21413 from the (local) times 5 min to 65 min. Also shown for comparison is the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing Fq in (3.4) with parameters A=6×107 and q=1.25.Figure 5
DART 21413: Data showingtsunami about 90 minutes after the earthquake and propagation model.Figure6 shows the actual run-up profile taken from the tide gauge at Wake Island at an even later time and greater distance. This is compared with our predicted run-up profile generated by (1.3) and (1.4) together with (1.5) for the forcing Fq in (3.4) with parameters A=6×107 and q=1.25. The run-up data and theoretical profile have similar profiles initially, for the first few oscillations.Figure 6
Wake Island: Data showingtsunami about 125 minutes after the earthquake and run-up model for Wake Island shoal.


## 3.1.q-Advanced Tsunami Model

Consider the forced one-dimensional wave equation for the water-level functionψ [14, 22],
(3.1)∂t2ψ(t,x)-c02∂x2ψ(t,x)=F(t,x),∀t∈ℝ,x∈ℝ,ψ(t,x)=0,∀t<0,x∈ℝ,
where the condition on the right in (3.1) constitutes the boundary and initial conditions. The forcing is expected to satisfy the conditions, for fixed t*, x*∈ℝ,
(3.2)lim|t|→∞F(t,x*)=0,lim|x|→∞F(t*,x)=0,∫-∞∞‍F(t*,x)dx=0,
and is related to the depth of the ocean floor by ∂t2H(t,x)=F(t,x). The models in [4, 8] start with ground-motion profiles H=HTS or H=HZWL where
(3.3)HTS≡Aγtanh(γx-tτ),HZWL≡(2Aγ)sech2(γx)tanh(γx)sin(πt2τ),
respectively. In [8], it is noted that using the first model H=HTS gives a force consistent with a landslide that continues for all time. In [4], the second model H=HZWL suggests an earthquake that continues for all time. In these settings, (3.2) does not hold. However, these models are still used as part of the initial forcings for the wave equation since they lead to integrable solutions whose evolutions resemble that of actual tsunamis. As a comparison, we propose the following q-advanced model:
(3.4)Hq(t,x)≡A·Kq(tτ)Sin(γ·x)q,Fq(t,x)≡∂2Hq∂t2=A·qτ2Kq(q2tτ)Sin(γ·x)q,
where Fq(t,x) now satisfies the conditions in (3.2). When (3.4) is substituted into (3.1), solved and simplified, one obtains a unique solution ψ(t,x) to the forced wave equation. To express the solution, define the two phase functions, corresponding to right and left propagation,
(3.5)φμ+(t,x,ξ)≡tτ-γ·x-ξqμ,φμ-(t,x,ξ)≡tτ+γ·x-ξqμ,∀t,x,ξ∈ℝ,
where the parameters are related by c0γτ=qμ. Define, for any α,β,μ∈ℝ and q>1,
(3.6)𝒯α,β,μ±(t,x)≡∫ξ=0γ·x‍Sin(qα·ξ)q·Kq(qβ·φμ±(t,x,ξ))dξ.
Then, for t>0 and x∈ℝ, the reader can verify that
(3.7)ψ(t,x)=ψparticular(t,x)+ψhomogeneous(t,x),
where a particular solution to (3.1), with the forcing in (3.4), is
(3.8)ψparticular(t,x)=(-A·q-μ2)·[𝒯0,1,μ-(t,x)-𝒯0,1,μ+(t,x)].
A smooth solution that satisfies ψ(t,x)=0, for t≤0, is obtained by choosing
(3.9)ψhom.(t,x)=-12[ψpar.(0,x+c0t)+ψpar.(0,x-c0t)+∫x-c0tx+c0t‍∂ψpar.(0,u)c0·∂tdu].
One can see from (3.5)–(3.9) that the solution has the basic wavelet properties
(3.10)ψ(·,x*),ψ(t*,·)∈ℒ1(ℝ)∩ℒ2(ℝ)∩ℒ∞(ℝ),∫ℝ‍ψ(t,x*)dt=∫ℝ‍ψ(t*,x)dx=0,
for each t*,x*∈ℝ. To analyze the long-term behavior of the solution ψ, we use (2.1), (2.17), and integration by parts, twice, to obtain Proposition 3.1.Proposition 3.1.
For anyα,β,μ∈ℝ and q>1 the following identity holds:
(3.11)𝒯α,β,μ±(t,x)=q-α·Kq(qβ·[tτ∓γxqμ])-q-2α+2β-2μ+1·𝒯α-1,β+2,μ±(t,x)-q-α·Cos(qαγx)q·Kq(qβtτ)±q-2α+β-μSinq(qα-1γx)·Kq(qβ+1tτ).The expression in (3.11) demonstrates that 𝒯α,β,μ±, as used in (3.8), can be written as a series of localized (bound) states and traveling (free) states. These different terms appear as translated and scaled versions of the wavelets Kq, Cosq and Sinq. Thus, such functions provide a good match for a wave profile ψ that was generated by the family of forcings Fq in (3.4). Furthermore, for t≫0, x≫0 and μ≫0 (reasonable for tsunamis),
(3.12)ψparticular(t,x)≃(Aq-3μ-12)·Kq(γ·q-μ[c0t-x]).
Finally, observe that Fq(t,x) in (3.4) is q-advanced in time compared to the ground-motion profile Hq(t,x). We also find that the forcing precedes the response ψ for the q-advanced models. Figure 4(a) illustrates a case where a forcing profile generates a similarly shaped tsunami in Figure 4(b).Remark 3.2.
Similar results hold for height functionsH~q(t,x)≡A~·Kq(t/τ)Cos(γ·x)q.


## 3.2. Numerical Solution of aq-Advanced Tsunami Wave Event

Here, we model the Japan tsunami of March 11, 2011, using (1.3), (1.4), and (1.5) with a forcing term Fq in (3.4) with environmental parameters chosen to be, in mks units,
(3.13)g0=10ms-2,H0=4000m,c0=200ms-1,γ=0.01m-1,τ=100s,A=6×107m,
which are based on past experience [4]. The sea-depth function is chosen to be
(3.14)H(x)=H0,forx≤0,H(x)=H0·[1-(x5km)2],forx∈(0,5km),
which models the sea-floor near Wake Island. This is a required modification of the model used in [7]. For stability, we employ a Lax-Wendroff correction term in the numerical method. A good match with typical tsunami profiles and run-up profiles was obtained by using q=1.25. In this case, μ≃23.74, which justifies the approximation in (3.12).Figure3 represents data from the Japan tsunami of March 11, 2011, at the first oceanic observation site DART 21418. This data includes a preliminary forcing profile from the (local) times 5 min to 20 min, along with the actual tsunami profile from the (local) times 42 min to 72 min.Figure 3
DART 21418: Data showing earthquake (forcing, on the left) and resulting tsunami (on the right) observed on March 11, 2011, from 5:43 am to 6:51 am (horizontal axis is time in minutes, vertical axis is in meters).(a) Comparison offorcing from data (dashed line) and model (solid line) results in q=1.25; (b) comparing tsunami at DART 21418 and propagation model results in A=6×107m.
(a)
(b)Figure4(a) again shows the Dart 21418 forcing profile from the times 5 min to 20 min, along with a q-advanced forcing profile as in (3.4), with A and q chosen to be A=6×107 and q=1.25 to effect comparable forcing profiles.Figure4(b) shows the Dart 21418 actual tsunami profile from the times 42 min to 72 min along with the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing Fq in (3.4) as above (with parameters A=6×107 and q=1.25).Figure5 shows the actual tsunami profile at a later time and greater distance at oceanic observation DART 21413 from the (local) times 5 min to 65 min. Also shown for comparison is the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing Fq in (3.4) with parameters A=6×107 and q=1.25.Figure 5
DART 21413: Data showingtsunami about 90 minutes after the earthquake and propagation model.Figure6 shows the actual run-up profile taken from the tide gauge at Wake Island at an even later time and greater distance. This is compared with our predicted run-up profile generated by (1.3) and (1.4) together with (1.5) for the forcing Fq in (3.4) with parameters A=6×107 and q=1.25. The run-up data and theoretical profile have similar profiles initially, for the first few oscillations.Figure 6
Wake Island: Data showingtsunami about 125 minutes after the earthquake and run-up model for Wake Island shoal.


## 4. Rogue-Wave Modeling Using Solutions of MADEs

A debate continues on the physical cause of rogue waves, [9]. One possible mechanism is a natural outcome of a constructive interference between rippling surface waves that propagate in different directions [22, page 191]. We construct a localized plane wave using a MADE-generated solution of the wave equation, for A>0,
(4.1)Ψq(t,x)=A·Cosq(tτ)·Cos(γ·x)q.
Substituting Ψq into (3.1) results in a small but persistent forcing. The model in (4.1) ignores possible translational behavior, which will be discussed at the end of this section. Here, γ and τ are parameters that satisfy cγ=1/τ for a constant celerity c>0. The forcing required to obtain Ψq from a calm distant past consists of two terms
(4.2)Fq(t,x)=Aqγ2c2·[Cosq(q·tτ)·Cos(γ·x)q-Cosq(tτ)·Cos(q·γ·x)q],
and it is easily seen that ∥Fq∥∞≤A/τ2. We show that ∥Fq∥∞→0 as q→1+ while from (4.1) it follows that ∥ψq∥∞=A remains constant for all q>1. This is a type of triad resonance with peak at (t,x)=(0,0). To proceed with our analysis, estimates on the differences between q-advanced functions, which are solutions of MADEs, are needed.


### 4.1. Analysis of Forcing Terms forqCos and qSin-Type Rogue Waves

In this section, we show that small amplitude forces, over long periods of time, can naturally produce large rogue waves. This demonstrates the existence of a resonance for the system externally forced by (4.2).Proposition 4.1.
Letϵ>0 be given. Then there exists QCϵ>1 such that for all q with 1<q<QCϵ one has
(4.3)|Cos(qt)q-Cos(t)q|<ϵ,
for all t∈ℝ; also there exists QSϵ>1 such that for all q with 1<q<QSϵ one has
(4.4)|Sin(qt)q-Sin(t)q|<ϵ,
for all t∈ℝ.The proof of this result is given in the last section. It is used here to show that small amplitude forces, over long periods of time, can naturally produce rogue waves. That is, forq sufficiently near 1, we now apply Proposition 4.1 to show that for arbitrarily small forcing terms Fq(t,x) there are large rogue solutions Ψq(t,x) of the forced wave equation (4.5) (and of (4.10)).Theorem 4.2.
LetA,c>0 and define Ψq as in (4.1) and Fq as in (4.2). Let ϵ>0 be given, and let q>1 be sufficiently close to 1. Then Ψq(t,x) satisfies the forced wave equation
(4.5)(∂t2-c2∂x2)Ψq(t,x)=Fq(t,x),
where Ψq(0,0)=A is fixed, but |Fq(t,x)|<ϵ for all t,x∈ℝ.Proof.
Without loss of generality, setγ=1 and c=1/τ. Then one has, for all q>1, that
(4.6)(∂t2-c2∂x2)Ψq(t,x)=(∂t2-c2∂x2)[A·Cos(ct)q·Cos(x)q]=A(-q)·Cos(qct)qc2·Cos(x)q-c2A·Cos(ct)q(-q)·Cos(qx)q=-Aqc2[Cos(qct)q·Cos(x)q-Cos(ct)q·Cos(qx)q]=Fq(t,x),
giving (4.5). Choose q0>1. Now let ϵ~≡ϵ/[2Aq0c2], and let QCϵ~>1 be chosen so that for 1<q<QCϵ~ one has |Cos(qx)q-Cos(x)q|<ϵ~ by Proposition 4.1. Then, for all 1<q<min{QCϵ~,q0},
(4.7)|Fq(t,x)|=|-Aqc2[Cos(qct)q·Cos(x)q-Cos(ct)q·Cos(x)q]-Aqc2[Cos(ct)q·Cos(x)q-Cos(ct)q·Cos(qx)q]|≤Aqc2|Cos(x)q||[Cos(qct)q-Cos(ct)q]|+Aqc2|Cos(ct)q||[Cos(x)q-Cos(qx)q]|(4.8)≤Aqc21|Cos(qct)q-Cos(ct)q|+Aqc21|Cos(qx)q-Cos(x)q|≤Aqc2ϵ~+Aqc2ϵ~<ϵ2+ϵ2=ϵ,
where (4.8) follows from the fact that ∥Cosq∥∞=1. Now, since Cos(0)q=1, we have Ψq(0,0)≡A·Cos(0)q·Cos(0)q=A.Theorem 4.3.
LetA,c>0. Let ϵ>0 be given, and let q>1 be sufficiently close to 1. Define
(4.9)Φq(t,x)≡A·Sin(ct)q·Sin(x)q,Gq(t,x)≡-Aq2c2[Sin(qct)q·Sin(x)q-Sin(ct)q·Sin(qx)q].
Then Φq(t,x) satisfies the forced wave equation
(4.10)(∂t2-c2∂x2)Φq(t,x)=Gq(t,x),
where supt,x∈ℝΦq≈A≫ϵ, but |Gq(t,x)|<ϵ for all t,x∈ℝ.Proof.
From (4.9), one has for all q>1 that
(4.11)(∂t2-c2∂x2)Φq(t,x)=(∂t2-c2∂x2)[A·Sin(ct)q·Sin(x)q]=A(-q2)·Sin(qct)qc2·Sin(x)q-c2A·Sin(ct)q(-q2)·Sin(qx)q=-Aq2c2[Sin(qct)q·Sin(x)q-Sin(ct)q·Sin(qx)q]=Gq(t,x),
giving (4.10). Choose q0>1. Now let ϵ~≡ϵ/[2Aq02c2q0], and let QSϵ~>1 be chosen so that for 1<q<QSϵ~ one has |Sin(qx)q-Sin(x)q|<ϵ~ by Proposition 4.1. Then, for all 1<q<min{QSϵ~,q0},
(4.12)|Gq(t,x)|=|-Aq2c2[Sin(qct)q·Sin(x)q-Sin(ct)q·Sin(x)q]+-Aq2c2[Sin(ct)q·Sin(x)q-Sin(ct)q·Sin(qx)q]|(4.13)≤Aq2c2|Sin(x)q||[Sin(qct)q-Sin(ct)q]|+Aq2c2|Sin(ct)q||[Sin(x)q-Sin(qx)q]|(4.14)≤Aq2c2q|Sin(qct)q-Sin(ct)q|+Aq2c2q|Sin(qx)q-Sin(x)q|(4.15)≤Aq2c2qϵ~+Aqc2qϵ~<ϵ2+ϵ2=ϵ,
where (4.14) follows from the fact that ∥Sinq∥∞≤q, as in [20]. Now, since supSinq≈1 for q>1 sufficiently close to 1, we have that supΦq≈A.


### 4.2. Slowly Moving Rogue Waves

When a slight drift in the rogue-generating current is present, there may be a speedv≪c to the wave-height profile. A model for such a rogue can be given by
(4.16)ψq,v(t,x)=A·Cos(Γv·γ·ct)q·Cos(γ·(x-vt))q,
where Γv≡1-(v/c)2. The peak of this wave still occurs at (t,x)=(0,0) but moves to the right at speed v. The techniques used in the previous section show that by choice of parameters, a small amplitude forcing, over a long period, will create a moving rogue of arbitrary size.Theorem 4.4.
LetA,c,γ>0 and c>v>0. Define Γv≡1-(v/c)2. Let ϵ>0 be given, and let q>1 be sufficiently close to 1. Define the surface-height function
(4.17)Ψ~q,v(t,x)≡A·Cos(Γv·γ·ct)q·Cos(γ·(x-vt))q,
and let the forcing be given by
(4.18)F~q(t,x)≡-(2Aγ2vcΓv)·Sin(Γvγct)qSin(γ(x-vct))q-Aγ2q(c2-v2)·[Cos(qΓvγct)qCos(γ(x-vt))q-Aγ2q(c2-v2)·+Cos(Γvγct)qCos(qγ(x-vt))q].
Then Ψ~qv(t,x) satisfies the forced wave equation
(4.19)(∂t2-c2∂x2)Ψ~q,v(t,x)=F~q(t,x),
where Ψ~q,v(0,0)=A is fixed, but |F~q(t,x)|<ϵ+(2AqΓvγ2c)·v for all t,x∈ℝ.Proof.
Applying the operator(∂t2-c2∂x2) to (4.17) yields (4.19) for F~q in (4.18). The magnitude of the first term in (4.18) is handled as follows:
(4.20)|(2Aγ2vcΓv)Sin(Γvγct)qSin(γ(x-vt))q|≤(2Aγ2vcΓv)|Sin(Γvγct)q|·|Sin(γ(x-vt))q|≤(2Aγ2vcΓv)qq=(2Aγ2qcΓv)·v,
where the last inequality follows from the fact that ∥Sinq∥∞≤q. The remaining expressions in (4.18) are controlled by noticing that
(4.21)|Cos(q·Γvγct)q·Cos(γ(x-vt))q-Cos(Γvγct)q·Cos(q·γ(x-vt))q|
can be brought below ϵ for q sufficiently close to 1+ by paralleling steps (4.7) through (4.8) in Theorem 4.2 and applying Proposition 4.1. The result is now proven.Remark 4.5.
Forv sufficiently small (as well as q sufficiently near 1+), the term (2AqΓvγ2c)·v in Theorem 4.4 is small, and one then has that F~q(t,x) can be made arbitrarily small compared with the rogue amplitude A, independently of t,x∈ℝ. For smaller values of γ, the moving rogue wave maintains a large amplitude near A for a longer period of time.Remark 4.6.
There is a corresponding theorem for the movingSinq rogue wave given by Φ~q,v(t,x)≡A·Sin(Γvγct)q·Sin(γ(x-vt))q.Remark 4.7.
Note that, as is demonstrated in Figure7, classic rogue wave profiles emerge from smaller forcings even for q relatively far from 1.Parameters:c=1.0, λ=1.0, τ=q=3/2, and γ=1/q=2/3. (a) Forcing profile Fq converges toward x=0 as t→0-; (b) rogue solution Ψq in a neighborhood of x=0 that oscillates over time; (c) forcing with ∥Fq∥∞≤1/q2=4/9 and solution with ∥Ψq∥∞=1.
(a)
(b)
(c)


## 4.1. Analysis of Forcing Terms forqCos and qSin-Type Rogue Waves

In this section, we show that small amplitude forces, over long periods of time, can naturally produce large rogue waves. This demonstrates the existence of a resonance for the system externally forced by (4.2).Proposition 4.1.
Letϵ>0 be given. Then there exists QCϵ>1 such that for all q with 1<q<QCϵ one has
(4.3)|Cos(qt)q-Cos(t)q|<ϵ,
for all t∈ℝ; also there exists QSϵ>1 such that for all q with 1<q<QSϵ one has
(4.4)|Sin(qt)q-Sin(t)q|<ϵ,
for all t∈ℝ.The proof of this result is given in the last section. It is used here to show that small amplitude forces, over long periods of time, can naturally produce rogue waves. That is, forq sufficiently near 1, we now apply Proposition 4.1 to show that for arbitrarily small forcing terms Fq(t,x) there are large rogue solutions Ψq(t,x) of the forced wave equation (4.5) (and of (4.10)).Theorem 4.2.
LetA,c>0 and define Ψq as in (4.1) and Fq as in (4.2). Let ϵ>0 be given, and let q>1 be sufficiently close to 1. Then Ψq(t,x) satisfies the forced wave equation
(4.5)(∂t2-c2∂x2)Ψq(t,x)=Fq(t,x),
where Ψq(0,0)=A is fixed, but |Fq(t,x)|<ϵ for all t,x∈ℝ.Proof.
Without loss of generality, setγ=1 and c=1/τ. Then one has, for all q>1, that
(4.6)(∂t2-c2∂x2)Ψq(t,x)=(∂t2-c2∂x2)[A·Cos(ct)q·Cos(x)q]=A(-q)·Cos(qct)qc2·Cos(x)q-c2A·Cos(ct)q(-q)·Cos(qx)q=-Aqc2[Cos(qct)q·Cos(x)q-Cos(ct)q·Cos(qx)q]=Fq(t,x),
giving (4.5). Choose q0>1. Now let ϵ~≡ϵ/[2Aq0c2], and let QCϵ~>1 be chosen so that for 1<q<QCϵ~ one has |Cos(qx)q-Cos(x)q|<ϵ~ by Proposition 4.1. Then, for all 1<q<min{QCϵ~,q0},
(4.7)|Fq(t,x)|=|-Aqc2[Cos(qct)q·Cos(x)q-Cos(ct)q·Cos(x)q]-Aqc2[Cos(ct)q·Cos(x)q-Cos(ct)q·Cos(qx)q]|≤Aqc2|Cos(x)q||[Cos(qct)q-Cos(ct)q]|+Aqc2|Cos(ct)q||[Cos(x)q-Cos(qx)q]|(4.8)≤Aqc21|Cos(qct)q-Cos(ct)q|+Aqc21|Cos(qx)q-Cos(x)q|≤Aqc2ϵ~+Aqc2ϵ~<ϵ2+ϵ2=ϵ,
where (4.8) follows from the fact that ∥Cosq∥∞=1. Now, since Cos(0)q=1, we have Ψq(0,0)≡A·Cos(0)q·Cos(0)q=A.Theorem 4.3.
LetA,c>0. Let ϵ>0 be given, and let q>1 be sufficiently close to 1. Define
(4.9)Φq(t,x)≡A·Sin(ct)q·Sin(x)q,Gq(t,x)≡-Aq2c2[Sin(qct)q·Sin(x)q-Sin(ct)q·Sin(qx)q].
Then Φq(t,x) satisfies the forced wave equation
(4.10)(∂t2-c2∂x2)Φq(t,x)=Gq(t,x),
where supt,x∈ℝΦq≈A≫ϵ, but |Gq(t,x)|<ϵ for all t,x∈ℝ.Proof.
From (4.9), one has for all q>1 that
(4.11)(∂t2-c2∂x2)Φq(t,x)=(∂t2-c2∂x2)[A·Sin(ct)q·Sin(x)q]=A(-q2)·Sin(qct)qc2·Sin(x)q-c2A·Sin(ct)q(-q2)·Sin(qx)q=-Aq2c2[Sin(qct)q·Sin(x)q-Sin(ct)q·Sin(qx)q]=Gq(t,x),
giving (4.10). Choose q0>1. Now let ϵ~≡ϵ/[2Aq02c2q0], and let QSϵ~>1 be chosen so that for 1<q<QSϵ~ one has |Sin(qx)q-Sin(x)q|<ϵ~ by Proposition 4.1. Then, for all 1<q<min{QSϵ~,q0},
(4.12)|Gq(t,x)|=|-Aq2c2[Sin(qct)q·Sin(x)q-Sin(ct)q·Sin(x)q]+-Aq2c2[Sin(ct)q·Sin(x)q-Sin(ct)q·Sin(qx)q]|(4.13)≤Aq2c2|Sin(x)q||[Sin(qct)q-Sin(ct)q]|+Aq2c2|Sin(ct)q||[Sin(x)q-Sin(qx)q]|(4.14)≤Aq2c2q|Sin(qct)q-Sin(ct)q|+Aq2c2q|Sin(qx)q-Sin(x)q|(4.15)≤Aq2c2qϵ~+Aqc2qϵ~<ϵ2+ϵ2=ϵ,
where (4.14) follows from the fact that ∥Sinq∥∞≤q, as in [20]. Now, since supSinq≈1 for q>1 sufficiently close to 1, we have that supΦq≈A.


## 4.2. Slowly Moving Rogue Waves

When a slight drift in the rogue-generating current is present, there may be a speedv≪c to the wave-height profile. A model for such a rogue can be given by
(4.16)ψq,v(t,x)=A·Cos(Γv·γ·ct)q·Cos(γ·(x-vt))q,
where Γv≡1-(v/c)2. The peak of this wave still occurs at (t,x)=(0,0) but moves to the right at speed v. The techniques used in the previous section show that by choice of parameters, a small amplitude forcing, over a long period, will create a moving rogue of arbitrary size.Theorem 4.4.
LetA,c,γ>0 and c>v>0. Define Γv≡1-(v/c)2. Let ϵ>0 be given, and let q>1 be sufficiently close to 1. Define the surface-height function
(4.17)Ψ~q,v(t,x)≡A·Cos(Γv·γ·ct)q·Cos(γ·(x-vt))q,
and let the forcing be given by
(4.18)F~q(t,x)≡-(2Aγ2vcΓv)·Sin(Γvγct)qSin(γ(x-vct))q-Aγ2q(c2-v2)·[Cos(qΓvγct)qCos(γ(x-vt))q-Aγ2q(c2-v2)·+Cos(Γvγct)qCos(qγ(x-vt))q].
Then Ψ~qv(t,x) satisfies the forced wave equation
(4.19)(∂t2-c2∂x2)Ψ~q,v(t,x)=F~q(t,x),
where Ψ~q,v(0,0)=A is fixed, but |F~q(t,x)|<ϵ+(2AqΓvγ2c)·v for all t,x∈ℝ.Proof.
Applying the operator(∂t2-c2∂x2) to (4.17) yields (4.19) for F~q in (4.18). The magnitude of the first term in (4.18) is handled as follows:
(4.20)|(2Aγ2vcΓv)Sin(Γvγct)qSin(γ(x-vt))q|≤(2Aγ2vcΓv)|Sin(Γvγct)q|·|Sin(γ(x-vt))q|≤(2Aγ2vcΓv)qq=(2Aγ2qcΓv)·v,
where the last inequality follows from the fact that ∥Sinq∥∞≤q. The remaining expressions in (4.18) are controlled by noticing that
(4.21)|Cos(q·Γvγct)q·Cos(γ(x-vt))q-Cos(Γvγct)q·Cos(q·γ(x-vt))q|
can be brought below ϵ for q sufficiently close to 1+ by paralleling steps (4.7) through (4.8) in Theorem 4.2 and applying Proposition 4.1. The result is now proven.Remark 4.5.
Forv sufficiently small (as well as q sufficiently near 1+), the term (2AqΓvγ2c)·v in Theorem 4.4 is small, and one then has that F~q(t,x) can be made arbitrarily small compared with the rogue amplitude A, independently of t,x∈ℝ. For smaller values of γ, the moving rogue wave maintains a large amplitude near A for a longer period of time.Remark 4.6.
There is a corresponding theorem for the movingSinq rogue wave given by Φ~q,v(t,x)≡A·Sin(Γvγct)q·Sin(γ(x-vt))q.Remark 4.7.
Note that, as is demonstrated in Figure7, classic rogue wave profiles emerge from smaller forcings even for q relatively far from 1.Parameters:c=1.0, λ=1.0, τ=q=3/2, and γ=1/q=2/3. (a) Forcing profile Fq converges toward x=0 as t→0-; (b) rogue solution Ψq in a neighborhood of x=0 that oscillates over time; (c) forcing with ∥Fq∥∞≤1/q2=4/9 and solution with ∥Ψq∥∞=1.
(a)
(b)
(c)


## 5. Wavelet Signal Analysis, Inversion, and the Frame Operator

We now have a collection of solutions to differential equations that give the qualitative behavior of a physical phenomena. Next, to detect, analyze, store, and recover a tsunami waveform, it is common to use a wavelet analysis [25]. The process begins by identifying a discrete set of functions, called an affine frame,
(5.1)Λψ≡{ψj,k(t)≡qj/2ψ(qjt-kb)∥ψ∥|j,k∈ℤ}.
With some effort, it can be shown that spanΛψ=ℒ2(ℝ), for appropriate ψ and b>0 sufficiently small [26]. This leads a wavelet transform Wψ:ℒ2(ℝ)→ℓ2(ℤ2) where
(5.2)Wψ[f](j,k)≡〈ψj,k,f〉=[qj/2∥ψ∥]∫ℝ‍ψ(qjt-kb)f(t)dt.
The range of W is a subspace of ℓ2(ℤ2) and has an adjoint W* defined to be
(5.3)W*:W[ℒ2(ℝ)]⊆ℓ2(ℤ2)→ℒ2(ℝ),W*[{cj,k}ℤ2]=∑‍cj,kψj,k*(t),
where ψj,k* are the elements of the dual frame to Λψ. The frame operator S≡W*W:ℒ2(ℝ)→ℒ2(ℝ) is invertible for b>0 sufficiently small. If Λψ is an orthonormal basis for ℒ2(ℝ), then one can use ψj,k*=ψj,k. The wavelets Kq, Cosq, and Sinq generate frames whose inner product structures are nearly orthogonal [20]. Consequently, as will be shown below, a reasonable analysis of the different waveforms is obtained without computing the dual to Λψ.The results of a signal analysis and synthesis, briefly presented here, consist of(1) choosing 256 equallyspaced points from the data, (2) computing the inner products with elements of Λψ, (3) sorting and identifying inner products with the largest magnitudes, (4) reconstructing the wavelet-based waveform, (5) normalizing in ℒ2, and (6) computing the normalized RMS error.By [19], the sizes chosen for the parameters are q=1.5 and b~2, with slight adjustments being made to improve the result. The reconstructed signal has an amplitude about b/(2q) of the size of the data profile. The need for this scaling was explained in [20] and is due to the fact that the ℒ2 operator norm of S-1 is estimated to be b/(2q).


### 5.1. Tsunami Wavelet Analysis

On March 11, 2011, an earthquake of magnitude 9.0 occurred off the coast of Japan causing a tsunami no less than10m high. Surface wave levels were detected by buoys operated by DART. To detect, analyze, store, and recover a tsunami waveform, we choose the wavelet frame
(5.4)ΛKq={qjCK·Kq(qjt-kb)|j,k∈ℤ},CK≡∫0∞‍Kq2(t)dt,
with q=1.5 and b=1.95. Then the application of WKq on tsunami data gives coefficients {cj,k} which can be plotted in a k versus j diagram. The largest values of |cj,k| indicate position (due to k) and narrowness (due to j). Only the largest 20 coefficients in magnitude, for the ranges -1≤j≤3 and 1≤k≤10, were used to obtain the fit on the bottom right of Figure 8. The normalized-RMS error
(5.5)NRMSE≡1max(yiwavelet)-min(yiwavelet)·(1N∑i=1N‍(yidata-yiwavelet)2)1/2
was computed to be NRMSE=16%. The bubble plots of the relative sizes of the coefficients are shown with the discrete time translation variable k displayed horizontally, and the discrete frequency exponentj displayed vertically.(a) Tsunami 46411 from DART, March 2011; (b) relative magnitude of coefficients for 3 scales{q-1,1,q} for q=1.5; (c) relative magnitude of coefficients for 5 scales {qj}j=-13; (d) Kq(t) wavelet approximation with 20 largest coefficients; (e) largest magnitude of Kq (q=1.5, scale = q0=1, shift = 2·b = 3.9); (f) comparison between DART data and Kq wavelet reconstruction.
(a)
(b)
(c)
(d)
(e)
(f)


### 5.2. Rogue Wavelet Analysis

On January 1, 1995, a rogue wave was detected on the Draupner platform in the North Sea. Surface wave heights were recorded using a laser-detection method [10]. Here we use the wavelet frame(5.6)ΛCosq={qjCC·Cosq(qjt-kb)∣j,k∈ℤ},CC≡∫0∞‍Cosq2(t)dt,
with q=1.5 and b=2.44. Here, we used 30 coefficients, for the ranges 1≤j≤4 and 1≤k≤10, to obtain the fit on the bottom right. The normalized RMS error is 18%. See Figure 9 for the results of this analysis.(a) Rogue 1520 from Draupner, January 1995; (b) relative magnitude of coefficients for 3 scales{1, q, q2} for q=1.5; (c) relative magnitude of coefficients for 5 scales {qj}j=04; (d) Cosq wavelet approximation using largest 30 coefficients; (e) largest magnitude of Cosq (q=1.5, scale = q2, shift = 5·b =12.2); (f) comparison between Draupner data and Cosq wavelet reconstruction.
(a)
(b)
(c)
(d)
(e)
(f)


## 5.1. Tsunami Wavelet Analysis

On March 11, 2011, an earthquake of magnitude 9.0 occurred off the coast of Japan causing a tsunami no less than10m high. Surface wave levels were detected by buoys operated by DART. To detect, analyze, store, and recover a tsunami waveform, we choose the wavelet frame
(5.4)ΛKq={qjCK·Kq(qjt-kb)|j,k∈ℤ},CK≡∫0∞‍Kq2(t)dt,
with q=1.5 and b=1.95. Then the application of WKq on tsunami data gives coefficients {cj,k} which can be plotted in a k versus j diagram. The largest values of |cj,k| indicate position (due to k) and narrowness (due to j). Only the largest 20 coefficients in magnitude, for the ranges -1≤j≤3 and 1≤k≤10, were used to obtain the fit on the bottom right of Figure 8. The normalized-RMS error
(5.5)NRMSE≡1max(yiwavelet)-min(yiwavelet)·(1N∑i=1N‍(yidata-yiwavelet)2)1/2
was computed to be NRMSE=16%. The bubble plots of the relative sizes of the coefficients are shown with the discrete time translation variable k displayed horizontally, and the discrete frequency exponentj displayed vertically.(a) Tsunami 46411 from DART, March 2011; (b) relative magnitude of coefficients for 3 scales{q-1,1,q} for q=1.5; (c) relative magnitude of coefficients for 5 scales {qj}j=-13; (d) Kq(t) wavelet approximation with 20 largest coefficients; (e) largest magnitude of Kq (q=1.5, scale = q0=1, shift = 2·b = 3.9); (f) comparison between DART data and Kq wavelet reconstruction.
(a)
(b)
(c)
(d)
(e)
(f)


## 5.2. Rogue Wavelet Analysis

On January 1, 1995, a rogue wave was detected on the Draupner platform in the North Sea. Surface wave heights were recorded using a laser-detection method [10]. Here we use the wavelet frame(5.6)ΛCosq={qjCC·Cosq(qjt-kb)∣j,k∈ℤ},CC≡∫0∞‍Cosq2(t)dt,
with q=1.5 and b=2.44. Here, we used 30 coefficients, for the ranges 1≤j≤4 and 1≤k≤10, to obtain the fit on the bottom right. The normalized RMS error is 18%. See Figure 9 for the results of this analysis.(a) Rogue 1520 from Draupner, January 1995; (b) relative magnitude of coefficients for 3 scales{1, q, q2} for q=1.5; (c) relative magnitude of coefficients for 5 scales {qj}j=04; (d) Cosq wavelet approximation using largest 30 coefficients; (e) largest magnitude of Cosq (q=1.5, scale = q2, shift = 5·b =12.2); (f) comparison between Draupner data and Cosq wavelet reconstruction.
(a)
(b)
(c)
(d)
(e)
(f)


## 6. Estimates for|Cos(qt)q-Cos(t)q| and |qSin(qt)-qSin(t)|

In this section, the estimates for the differences ofq-advanced trigonometric functions are proven. First we record useful Fourier transform expressions. From (2.3) we have the Fourier transforms of Cos(t)q,Cos(qt)q,Sin(t)q, and Sin(qt)q, respectively, are given by
(6.1)ℱ[Cos(t)q](ω)=2(μq2)3Nq2π1θ(q2;ω2),(6.2)ℱ[Cos(qt)q](ω)=1q2(μq2)3Nq2π1θ(q2;(q2)-1ω2)=ω2qℱ[Cos(t)q](ω),(6.3)ℱ[Sin(t)q](ω)=2(μq2)3Nq2π(-iω)θ(q2;ω2),(6.4)ℱ[Sin(qt)q](ω)=1q2(μq2)3Nq2π(-iω/q)θ(q2;(q2)-1ω2)=ω2q2ℱ[Sin(t)q](ω),
where the algebraic identity θ(q2;(q2)nω2)=(q2)n(n+1)/2(ω2)nθ(q2;ω2), that follows from (2.6), was used to obtain (6.2) and (6.4). Further details are presented in [21].Proof of Proposition4.1.
We first prove the estimate for the differences involvingCosq. From (6.1) and (6.2), one has
(6.5)|Cos(qt)q-Cos(t)q|=|ℱ-1(ℱ[Cos(qt)q])-ℱ-1(ℱ[Cos(t)q])|=12π|∫ℝ‍eiωt{1q2(μq2)3Nq2πω2θ(q2;ω2)12π+++∫ℝ‍eiωt-2(μq2)3Nq2π1θ(q2;ω2)}dω|(6.6)≤12π1q2(μq2)3Nq2π∫ℝ‍|ω2-q|θ(q2;ω2)dω=2(μq2)3Nq2πq2∫0∞‍|ω2-q|θ(q2;ω2)dω,(6.7)=2πq(μq2)3Nq[∫01‍|ω2-q|θ(q2;ω2)dω+∫1∞‍|ω2-q|θ(q2;ω2)dω],
where the triangle inequality gives the inequality in (6.6), and the evenness of |ω2-q|/θ(q2;ω2) allows for reduction to integration over [0,∞) in (6.6).
The change of variablesω=1/u is made on the first integral in (6.7), and the algebraic identity θ(q2;u-2)=u-2θ(q2;u2) is used to obtain
(6.8)∫01‍|ω2-q|θ(q2;ω2)dω=∫∞1‍|u-2-q|θ(q2;u-2)(-u-2)du=∫1∞‍|u-2-q|θ(q2;u2)du.
Now (6.8) is used to reexpress the bound (6.7) as
(6.9)|Cos(qt)q-Cos(t)q|≤2πq(μq2)3Nq∫1∞‍|ω-2-q|+|ω2-q|θ(q2;ω2)dω.
From (2.9), we have that
(6.10)1θ(q2;ω2)≤[q1/4|ω|e(ln(|ω|))2/ln(q){πln(q)-1}]-1=ln(q)[1ln(q)q-1/4|ω|-1e-(ln(|ω|))2/ln(q){π-ln(q)}],
for 1<q<eπ. Deploying the bound (6.10) within the integral in the bound (6.9) gives
(6.11)|Cos(qt)q-Cos(t)q|≤F(q)[1ln(q)∫1∞‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω],
where
(6.12)F(q)≡2ln(q)(μq2)3Nqπq5/41π-ln(q).
It follows from [21] that
(6.13)limq→1+[ln(q)(μq2)3Nq]=π2⇒limq→1+F(q)=1π.
We now show that|Cos(qt)q-Cos(t)q| can be made arbitrarily small, independently of t, for all q>1 sufficiently close to 1+. In light of (6.13), this is accomplished by first showing the corresponding statement holds for the bracketed expression in (6.11).
Letα>0 be arbitrary, with α to be specified later. The integral in ω over the interval [1,∞) in (6.11) is now subdivided into two integrals, the first over [1,eαln(q)] and the second over [eαln(q),∞). First, considering ω restricted to the interval [1,eαln(q)], one has that
(6.14)1ln(q)∫1eαln(q)‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω≤1ln(q)∫1eαln(q)‍{|ω-2-q|ω2+|ω2-q|}ω-1e(lnω)2/ln(q)dω=1ln(q)∫1eαln(q)‍{|ω-1-qω|+|ω-qω-1|}e(lnω)2/ln(q)dω.
Now the function |ω-1-qω| assumes its maximum value on [1,eαln(q)] at the right endpoint eαln(q). In addition, for all q sufficiently close to 1, the function |ω-qω-1| assumes its maximum value on [1,eαln(q)] at the right endpoint eαln(q). The latter statement follows since g(ω)=ω-qω-1 is increasing, and one needs  only to compare endpoints |g(1)|=q-1 and |g(eαln(q))|=eαln(q)-qe-αln(q) to determine the larger value. An application of L'Hopital's rule gives that limq→1+|g(eαln(q))|/|g(1)|=∞. Thus, for q near 1, on the interval [1,eαln(q)], one has
(6.15){|ω-1-qω|+|ω-qω-1|}≤|e-αln(q)-qeαln(q)|+|eαln(q)-qe-αln(q)|=qeαln(q)-e-αln(q)+eαln(q)-qe-αln(q)=(q+1)(eαln(q)-e-αln(q)).
Thus, we bound the integral in (6.14) by the length of the interval times the bound (6.15) on the numerator. This gives
(6.16)1ln(q)∫1eαln(q)‍{|ω-1-qω|+|ω-qω-1|}e(lnω)2/ln(q)dω≤1ln(q)∫1eαln(q)‍(q+1)(eαln(q)-e-αln(q))dω≤1ln(q)(q+1)(eαln(q)-e-αln(q))(eαln(q)-1).
An application of L'Hopital's rule gives that
(6.17)limq→1+(eαln(q)-e-αln(q))ln(q)=2α,
which implies that in (6.16) we have
(6.18)limq→1+[(q+1)(eαln(q)-e-αln(q))ln(q)(eαln(q)-1)]=2·2α·0=0.
Combining (6.18) with (6.16) and (6.14) gives that
(6.19)limq→1+[1ln(q)∫1eαln(q)‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω]=0.
We next estimate the portion of the integral in (6.11) over the interval [eαln(q),∞).
(6.20)1ln(q)∫eαln(q)∞‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω=1ln(q)∫eαln(q)∞‍{|ω-3-qω-1|+|ω-qω-1|}e(lnω)2/ln(q)dω≤1ln(q)∫eαln(q)∞‍{ω-3+2qω-1+ω}e(lnω)2/ln(q)dω,
by bounding the integrals of the summands in (6.20). For each exponent k∈ℤ one has
(6.21)1ln(q)∫eαln(q)∞‍ωke(lnω)2/ln(q)dω=1ln(q)∫eαln(q)∞‍eklnωe-(lnω)2/ln(q)dω=qk2/4ln(q)∫eαln(q)∞‍e-(1/ln(q))[lnω-kln(q)/2]2dω,
where a completion of squares gives (6.21). We use the following estimate from [21]
(6.22)∫C∞‍e-A[lnω+B]2dω≤π2e1/(4A)-BAe-A[ln(C)+B-1/(2A)]2,
which is applied to (6.21) with A=1/ln(q),B=-kln(q)/2, and C=eαln(q) to obtain
(6.23)qk2/4ln(q)∫eαln(q)∞‍e-(1/ln(q))[lnω-kln(q)/2]2dω≤qk2/4ln(q)π2eln(q)/4+kln(q)/21/ln(q)e-(1/ln(q))[ln(eαln(q))-kln(q)/2-ln(q)/2]2=q(k+1)2/4π2e-(1/ln(q))[αln(q))-(k+1)ln(q)/2]2=q(k+1)2/4π2e-α2+α(k+1)ln(q)-(k+1)2ln(q)/4=π2e-α2eα(k+1)ln(q).
Combining (6.21) and (6.23) gives that, for each k∈ℤ,
(6.24)1ln(q)∫eαln(q)∞‍ωke(lnω)2/ln(q)dω≤π2e-α2eα(k+1)ln(q).
Revisiting (6.20) and applying (6.24) now gives
(6.25)1ln(q)∫eαln(q)∞‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω(6.26)≤1ln(q)∫eαln(q)∞‍{ω-3+2qω-1+ω}e(lnω)2/ln(q)dω≤π2e-α2[eα(-2)ln(q)+2qeα(0)ln(q)+eα(2)ln(q)].
Since the expression in (6.26) approaches 2πe-α2 as q→1+, by choosing α sufficiently large one can make (6.26), and hence (6.25), arbitrarily small for q sufficiently close to 1.
Now, letϵ>0 be given. By (6.11) we have for all α>0(6.27)|Cos(qt)q-Cos(t)q|≤F(q)[1ln(q)∫1eαln(q)‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω]+F(q)[1ln(q)∫eαln(q)∞‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω].
One has from (6.13)
(6.28)limq→1+F(q)π2e-α2[eα(-2)ln(q)+2qeα(0)ln(q)+eα(2)ln(q)]=2e-α2.
Choose A~>2 and then fix α>0 such that A~e-α2<ϵ/2. By (6.26) and (6.28), there exists q1=q1(A~,α,ϵ) such that for all 1<q<q1 one has
(6.29)0<F(q)[1ln(q)∫eαln(q)∞‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω]≤F(q)π2e-α2[eα(-2)ln(q)+2qeα(0)ln(q)+eα(2)ln(q)]<A~e-α2<ϵ2.
For this value of α, by virtue of (6.19), one has
(6.30)limq→1+F(q)[1ln(q)∫1eαln(q)‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω]=1π·0=0,
which in turn says that there is a q2=q2(α,ϵ) such that for all 1<q<q2 one has
(6.31)0<F(q)[1ln(q)∫1eαln(q)‍{|ω-2-q|+|ω2-q|}ω-1e(lnω)2/ln(q)dω]<ϵ2.
Thus, for QCϵ=Q(A~,α,ϵ)=min{q1,q2}, and for all 1<q<QCϵ, applying (6.31) and (6.29) to (6.27) yields
(6.32)|Cos(qt)q-Cos(t)q|<ϵ2+ϵ2<ϵ,
independently of t∈ℝ. The proposition is now proven forCosq.
The proof forSinq parallels the above argument for Cosq, so we only outline it here, indicating points of slight differences. From (6.3) and (6.4), the analogue of (6.6) becomes (6.33) below
(6.33)|Sin(qt)q-Sin(t)q|=|ℱ-1(ℱ[Sin(qt)q])-ℱ-1(ℱ[Sin(t)q])|≤12π1q22(μq2)3Nq2π∫ℝ‍|ω3-q2ω|θ(q2;ω2)dω=2(μq2)3Nqπq2∫0∞‍|ω3-q2ω|θ(q2;ω2)dω,(6.34)=2πq2(μq2)3Nq∫1∞‍|ω-3-q2ω-1|+|ω3-q2ω|θ(q2;ω2)dω,
where a reciprocation change of variables on the integral on [0,1] converts to an integral on [1,∞), yielding (6.34) as the analogue of (6.9). Deploying the bound (6.10) to (6.34) now gives
(6.35)|Sin(qt)q-Sin(t)q|≤F(q)q[1ln(q)∫1∞‍{|ω-4-q2ω-2|+|ω2-q2|}e(lnω)2/ln(q)dω],
with F(q) defined in (6.12). For the integral in ω in (6.35) over the interval [eαln(q),∞), an application of (6.24) in conjunction with the triangle inequality lets us proceed directly to an analogue of (6.26), with key factor e-α2 still intact and the remaining factor approaching a constant as q→1+. On the other hand, the integral over the interval [1,eαln(q)] in (6.35) proceeds as follows. One has
(6.36)[1ln(q)∫1eαln(q)‍{|ω-4-q2ω-2|+|ω2-q2|}e(lnω)2/ln(q)dω]≤[1ln(q)∫1eαln(q)‍{|ω-4-q2ω-2|ω2+|ω2-q2|}dω]=[1ln(q)∫1eαln(q)‍{|ω-2-q2|+|ω2-q2|}dω]≤1ln(q){|e-2αln(q)-q2|+|e2αln(q)-q2|}[eαln(q)-1](6.37)=1ln(q){-e-2αln(q)+e2αln(q)}[eαln(q)-1],
where (6.36) and (6.37) hold for q sufficiently near 1+. Since limx→0[(-e-2αx+e2αx)/x]=4α, we have the limit as q→1+ of expression (6.37) is 0. The remainder of the proof is now entirely parallel to that of the Cosq case.



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*Source: 414060-2012-09-25.xml*