q -Advanced Models for Tsunami and Rogue Waves

Abstract

A wavelet [subscript] Kq[/subscript] (t ) , that satisfies the q-advanced differential equation [superscript] K q [variant prime][/superscript] ( t ) =[subscript] K q[/subscript] (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 [subscript] Fq[/subscript] ( t ,x ) =[subscript] Kq[/subscript] [subscript] (t )q[/subscript] Sin (x ) . The profile [subscript] F q[/subscript] is similar to tsunami models in present use. The function Sin [superscript] ( t ) [/superscript] q is a wavelet that satisfies a q-advanced harmonic oscillator equation. It is also shown that another wavelet, Cos [superscript] ( t ) [/superscript] 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 [subscript] K q[/subscript] and [superscript] Cos q [/superscript] is examined on actual data.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Ferhan M. Atici

Department of Mathematics, East Carolina University, Greenville, NC 27858, USA

Received 9 March 2012; Revised 19 May 2012; Accepted 21 May 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 _{K q} (x -ct ) where _{K q} 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 _{K q} -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 = _{K q} (t ) for q =1.5 .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

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) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Tsunami and rogue waves are perturbations of the water-surface elevation function η (t ,x ,y ) . The surface velocity components U ...1; (u ,v ,w ) are small and the time-averaged mean-flow velocity Y9;U YA; is assumed to remain 0 everywhere. Subject to assumptions, the functions _{K q} , ^{Cos q} and ^{Sin q} will be used in the construction of the forcing terms in the Matsuno equations [ 13] as given by [figure omitted; refer to PDF] where H (x ,y ) is the depth of the water, w ...1; ∂ η / ∂t is the vertical velocity of the wave surface, _{P x} and _{P y} are variable external forcings, and Q is a mass source term. The acceleration due to gravity _{g 0} accounts for buoyancy, and the Coriolis parameter _{f 0} addresses the rotation of the earth (only required for very long waves over the earth [ 14]). The friction coefficient α ...5;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 the x -direction. Thus, we set [figure omitted; refer to PDF] In deep water H = _{H 0} >0 is a fixed constant and _{c 0} = _{g 0 H 0} is a constant surface-wave speed (called the celerity). Then, for small amplitudes, the elevation function η (t ,x ) satisfies [figure omitted; refer to PDF] These equations are found in [ 8, Equation (2 ) ], [ 15, Equation (45 ) ], and [ 16, Equation (13 ) ], respectively. The new coefficients are defined as β (x ) ...1;3c / (4H ) and ν (x ) ...1;c ^{H 2} /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 forcings F (t ,x ) in ( 1.3) will be constructed from bounded wavelets [varphi] (u ) that satisfy MADEs. As such they satisfy the following conditions: [figure omitted; refer to PDF] Physically, if [varphi] 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 *} ∈ ... ), 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 forcing F is expressed in terms of _{K q} , ^{Cos q} and ^{Sin q} 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 _{K q} and ^{Cos q} . 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 function _{K q} (t ) that satisfies the MADE [figure omitted; refer to PDF] We set _{K q} (t ) ...1;0 for t ...4;0 , and define [figure omitted; refer to PDF] Note that _{K q} (t ) satisfies the wavelet conditions in ( 1.6), see [ 18]. Furthermore, as in [ 20], the reproducing kernel of _{K q} (t ) gives the functions [figure omitted; refer to PDF] which are displayed in Figure 2for q =1.5 . The normalization constant _{N q} is chosen so that Cos ^{( 0 )} q =1 . It is shown in [ 20] that the q -advanced harmonic oscillator equations, [figure omitted; refer to PDF] hold, which are second-order MADEs. Each of ( 2.2)-( 2.3) lies in ^{[Lagrangian (script capital L)] 2} . We show as a consequence of Theorem 2.1that Cos ^{( t )} q [arrow right]cos (t ) and Sin ^{( t )} q [arrow right]Sin (t ) pointwise for t ∈ ... as q [arrow right] ^{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 ^{[Lagrangian (script capital L)] 2} approximations of cos (t ) and sin (t ) , respectively, which are solutions of the limit of equations in ( 2.4) as q [arrow right] ^{1 +} .

2.1. Theta Functions

As part of our study, we employ the Jacobi theta function, defined for ω ∈ ... \ { - ^{q n} _{} n ∈ ...} , [figure omitted; refer to PDF] This function will only be used in regions where ω >0 and q >1 . Clearly the function θ ( ^{q 2} , ^{ω 2} ) is ^{C ∞} for all ω ∈ ... - {0 } and grows faster than any rational polynomial at ± ∞ and 0. As such, for each q >1 one has that 1 / θ ( ^{q 2} , ^{ω 2} ) is a Schwartz function that is flat at ω =0 . It also satisfies the algebraic identity [figure omitted; refer to PDF] which plays an important role in our analysis, [ 18].

Theorem 2.1.

For q >1 , let [figure omitted; refer to PDF] Then _{δ q} ( ω ) is a delta sequence in q > ^{1 +} , at ω =1 and has the following properties: [figure omitted; refer to PDF]

Proof.

From [ 21], for 1 <q < e π and for ω >0 one has [figure omitted; refer to PDF] Multiplication of ( 2.9) and ( 2.10) by ln (q ) , followed by reciprocation, gives that for ω >0 [figure omitted; refer to PDF] Now, (i ) follows from ( 2.11) and the fact that for k >0 , lim q [arrow right] 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 [figure omitted; refer to PDF] The functional identity ( μ q ) 4 q =2 || K q || 2 2 ( μ q 2 ) 2 N q , established in [ 20], allows for the replacement of ( || K q || 2 2 /q ) by ( ( μ q ) 4 / [2 ( μ q 2 ) 2 N q ] ) in ( 2.12) to obtain the following: [figure omitted; refer to PDF] This in turn gives that [figure omitted; refer to PDF] where the last equality follows from the q -Wallis limit, lim q [arrow right] 1 + [ln (q ) N q ( μ q 2 ) 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), [figure omitted; refer to PDF] In [ 20], we use the inverse Fourier transform to compute the expressions [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] which verify the MADEs in ( 2.4). Now Theorem 2.1gives that both _{δ q} ( ω ) and _{δ q} ( - 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 [figure omitted; refer to PDF] 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 ∈ ... and q >1 , [figure omitted; refer to PDF]

Remark 2.2.

The use of theta functions in frequency space provides decaying versions of cosine 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 speed _{c 0} = _{g 0 H 0} mainly depends on the average depth _{H 0} of the ocean. For tsunamis, the wavelength is much longer than _{H 0} 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 the y 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 function ψ [ 14, 22], [figure omitted; refer to PDF] 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 *} ∈ ... , [figure omitted; refer to PDF] and is related to the depth of the ocean floor by ^{∂ t 2} H (t ,x ) =F (t ,x ) . The models in [ 4, 8] start with ground-motion profiles H = _{H TS} or H = _{H ZWL} where [figure omitted; refer to PDF] respectively. In [ 8], it is noted that using the first model H = _{H TS} gives a force consistent with a landslide that continues for all time. In [ 4], the second model H = _{H ZWL} 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: [figure omitted; refer to PDF] where _{F q} (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, [figure omitted; refer to PDF] where the parameters are related by _{c 0} γ τ = ^{q μ} . Define, for any α , β , μ ∈ ... and q >1 , [figure omitted; refer to PDF] Then, for t >0 and x ∈ ... , the reader can verify that [figure omitted; refer to PDF] where a particular solution to ( 3.1), with the forcing in ( 3.4), is [figure omitted; refer to PDF] A smooth solution that satisfies ψ (t ,x ) =0 , for t ...4;0 , is obtained by choosing [figure omitted; refer to PDF] One can see from ( 3.5)-( 3.9) that the solution has the basic wavelet properties [figure omitted; refer to PDF] for each _{t *} , _{x *} ∈ ... . 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 any α , β , μ ∈ ... and q >1 the following identity holds: [figure omitted; refer to PDF]

The expression in ( 3.11) demonstrates that ^{...AF; α , 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 _{K q} , ^{Cos q} and ^{Sin q} . Thus, such functions provide a good match for a wave profile ψ that was generated by the family of forcings _{F q} in ( 3.4). Furthermore, for t ...b;0 , x ...b;0 and μ ...b;0 (reasonable for tsunamis), [figure omitted; refer to PDF] Finally, observe that _{F q} (t ,x ) in ( 3.4) is q -advanced in time compared to the ground-motion profile _{H q} (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 functions _{H ~ q} (t ,x ) ...1; A ~ · _{K q} ( t / τ ) Cos ( γ ·x ) q .

3.2. Numerical Solution of a q -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 _{F q} in ( 3.4) with environmental parameters chosen to be, in mks units, [figure omitted; refer to PDF] which are based on past experience [ 4]. The sea-depth function is chosen to be [figure omitted; refer to PDF] 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).

Figure 3represents 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).

[figure omitted; refer to PDF]

(a) Comparison of forcing 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 ×1 ^{0 7} m .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 4(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 ×1 ^{0 7} and q =1.25 to effect comparable forcing profiles.

Figure 4(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 _{F q} in ( 3.4) as above (with parameters A =6 ×1 ^{0 7} and q =1.25 ).

Figure 5shows 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 _{F q} in ( 3.4) with parameters A =6 ×1 ^{0 7} and q =1.25 .

Figure 5: DART 21413: Data showing tsunami about 90 minutes after the earthquake and propagation model.

[figure omitted; refer to PDF]

Figure 6shows 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 _{F q} in ( 3.4) with parameters A =6 ×1 ^{0 7} 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 showing tsunami about 125 minutes after the earthquake and run-up model for Wake Island shoal.

[figure omitted; refer to PDF]

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 , [figure omitted; refer to PDF] Substituting _{Ψ q} 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 c γ =1 / τ for a constant celerity c >0 . The forcing required to obtain _{Ψ q} from a calm distant past consists of two terms [figure omitted; refer to PDF] and it is easily seen that || _{F q || ∞} ...4;A / ^{τ 2} . We show that || _{F q || ∞} [arrow right]0 as q [arrow right] ^{1 +} while from ( 4.1) it follows that || _{ψ q || ∞} =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 for _q Cos and _q Sin -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.

Let ... >0 be given. Then there exists ^{Q C ...} >1 such that for all q with 1 <q < ^{Q C ...} one has [figure omitted; refer to PDF] for all t ∈ ... ; also there exists ^{Q S ...} >1 such that for all q with 1 <q < ^{Q S ...} one has [figure omitted; refer to PDF] for all t ∈ ... .

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, for q sufficiently near 1 , we now apply Proposition 4.1to show that for arbitrarily small forcing terms _{F q} (t ,x ) there are large rogue solutions _{Ψ q} (t ,x ) of the forced wave equation ( 4.5) (and of ( 4.10)).

Theorem 4.2.

Let A ,c >0 and define _{Ψ q} as in ( 4.1) and _{F q} as in ( 4.2). Let ... >0 be given, and let q >1 be sufficiently close to 1 . Then _{Ψ q} (t ,x ) satisfies the forced wave equation [figure omitted; refer to PDF] where _{Ψ q} (0,0 ) =A is fixed, but | _{F q} (t ,x ) | < ... for all t ,x ∈ ... .

Proof.

Without loss of generality, set γ =1 and c =1 / τ . Then one has, for all q >1 , that [figure omitted; refer to PDF] giving ( 4.5). Choose _{q 0} >1 . Now let ... ~ ...1; ... / [2A _{q 0} ^{c 2} ] , and let ^{Q C ... ~} >1 be chosen so that for 1 <q < ^{Q C ... ~} one has | Cos ^{( qx )} q - Cos ^{( x )} q | < ... ~ by Proposition 4.1. Then, for all 1 <q < min { ^{Q C ... ~} , _{q 0} } , [figure omitted; refer to PDF] where ( 4.8) follows from the fact that _{|| ^{Cos q} || ∞} =1 . Now, since Cos ^{( 0 )} q =1 , we have _{Ψ q} ( 0,0 ) ...1;A · Cos ^{( 0 )} q · Cos ^{( 0 )} q =A .

Theorem 4.3.

Let A ,c >0 . Let ... >0 be given, and let q >1 be sufficiently close to 1 . Define [figure omitted; refer to PDF] Then _{Φ q} (t ,x ) satisfies the forced wave equation [figure omitted; refer to PDF] where _{sup t ,x ∈ ... Φ q} [approximate]A ...b; ... , but | _{G q} (t ,x ) | < ... for all t ,x ∈ ... .

Proof.

From ( 4.9), one has for all q >1 that [figure omitted; refer to PDF] giving ( 4.10). Choose _{q 0} >1 . Now let ... ~ ...1; ... / [2A ^{_{q 0} 2 c 2} _{q 0} ] , and let ^{Q S ... ~} >1 be chosen so that for 1 <q < ^{Q S ... ~} one has | Sin ^{( qx )} q - Sin ^{( x )} q | < ... ~ by Proposition 4.1. Then, for all 1 <q < min { ^{Q S ... ~} , _{q 0} } , [figure omitted; refer to PDF] where ( 4.14) follows from the fact that _{|| ^{Sin q} || ∞} ...4; q , as in [ 20]. Now, since sup ^{Sin q} [approximate]1 for q >1 sufficiently close to 1 , we have that sup _{Φ q} [approximate]A .

4.2. Slowly Moving Rogue Waves

When a slight drift in the rogue-generating current is present, there may be a speed v ...a;c to the wave-height profile. A model for such a rogue can be given by [figure omitted; refer to PDF] where _{Γ v} ...1; 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.

Let A ,c , γ >0 and c >v >0 . Define _{Γ v} ...1; 1 - (v /c ^{) 2} . Let ... >0 be given, and let q >1 be sufficiently close to 1 . Define the surface-height function [figure omitted; refer to PDF] and let the forcing be given by [figure omitted; refer to PDF] Then _{Ψ ~ qv} (t ,x ) satisfies the forced wave equation [figure omitted; refer to PDF] where _{Ψ ~ q ,v} (0,0 ) =A is fixed, but | _{F ~ q} (t ,x ) | < ... + (2Aq _{Γ v} ^{γ 2} c ) ·v for all t ,x ∈ ... .

Proof.

Applying the operator ( ^{∂ t 2} - ^{c 2 ∂ x 2} ) 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: [figure omitted; refer to PDF] where the last inequality follows from the fact that _{|| ^{Sin q} || ∞} ...4; q . The remaining expressions in ( 4.18) are controlled by noticing that [figure omitted; refer to PDF] can be brought below ... for q sufficiently close to ^{1 +} by paralleling steps ( 4.7) through ( 4.8) in Theorem 4.2and applying Proposition 4.1. The result is now proven.

Remark 4.5.

For v sufficiently small (as well as q sufficiently near ^{1 +} ), the term (2Aq _{Γ v} ^{γ 2} c ) ·v in Theorem 4.4is 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 ∈ ... . 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 moving ^{Sin q} rogue wave given by _{Φ ~ q ,v} ( t ,x ) ...1;A · Sin ^{( _{Γ v} γct )} q · Sin ^{( γ ( x -vt ) )} q .

Remark 4.7.

Note that, as is demonstrated in Figure 7, 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 _{F q} converges toward x =0 as t [arrow right] ^{0 -} ; (b) rogue solution _{Ψ q} in a neighborhood of x =0 that oscillates over time; (c) forcing with || _{F q || ∞} ...4;1 / ^{q 2} =4 /9 and solution with || _{Ψ q || ∞} =1 .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

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, [figure omitted; refer to PDF] With some effort, it can be shown that span _{Λ ψ} = ^{[Lagrangian (script capital L)] 2} ( ... ) , for appropriate ψ and b >0 sufficiently small [ 26]. This leads a wavelet transform _{W ψ} : ^{[Lagrangian (script capital L)] 2} ( ... ) [arrow right] ^{[cursive l] 2} ( ^{... 2} ) where [figure omitted; refer to PDF] The range of W is a subspace of ^{[cursive l] 2} ( ^{... 2} ) and has an adjoint ^{W *} defined to be [figure omitted; refer to PDF] where ^{ψ j ,k *} are the elements of the dual frame to _{Λ ψ} . The frame operator S ...1; ^{W *} W : ^{[Lagrangian (script capital L)] 2} ( ... ) [arrow right] ^{[Lagrangian (script capital L)] 2} ( ... ) is invertible for b >0 sufficiently small. If _{Λ ψ} is an orthonormal basis for ^{[Lagrangian (script capital L)] 2} ( ... ) , then one can use ^{ψ j ,k *} = _{ψ j ,k} . The wavelets _{K q} , ^{Cos q} , and ^{Sin q} 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 _{Λ ψ} .

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 _{Λ ψ} , (3 ) sorting and identifying inner products with the largest magnitudes, (4 ) reconstructing the wavelet-based waveform, (5 ) normalizing in ^{[Lagrangian (script capital L)] 2} , 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 ^{[Lagrangian (script capital L)] 2} 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 than 10 m 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 [figure omitted; refer to PDF] with q =1.5 and b =1.95 . Then the application of _{W K q} on tsunami data gives coefficients { _{c j ,k} } which can be plotted in a k versus j diagram. The largest values of | _{c j ,k} | indicate position (due to k ) and narrowness (due to j ). Only the largest 20 coefficients in magnitude, for the ranges -1 ...4;j ...4;3 and 1 ...4;k ...4;10 , were used to obtain the fit on the bottom right of Figure 8. The normalized- RMS error [figure omitted; refer to PDF] 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 exponent j 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 { ^{q j } j = -1 3} ; (d) _{K q} (t ) wavelet approximation with 20 largest coefficients; (e) largest magnitude of _{K q} ( q =1.5 , scale = ^{q 0} =1 , shift = 2 ·b = 3.9); (f) comparison between DART data and _{K q} wavelet reconstruction.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

(f) [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] with q =1.5 and b =2.44 . Here, we used 30 coefficients, for the ranges 1 ...4;j ...4;4 and 1 ...4;k ...4;10 , to obtain the fit on the bottom right. The normalized RMS error is 18 % . See Figure 9for the results of this analysis.

(a) Rogue 1520 from Draupner, January 1995; (b) relative magnitude of coefficients for 3 scales {1 , q , ^{q 2} } for q =1.5 ; (c) relative magnitude of coefficients for 5 scales { ^{q j } j =0 4} ; (d) Cos q wavelet approximation using largest 30 coefficients; (e) largest magnitude of Cos q ( q =1.5 , scale = ^{q 2} , shift = 5 ·b =12.2); (f) comparison between Draupner data and Cos q wavelet reconstruction.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

(f) [figure omitted; refer to PDF]

6. Estimates for | Cos ( qt ) q - Cos (t ) q | and _{| q} Sin (qt ) _{- q} Sin (t ) |

In this section, the estimates for the differences of q -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 [figure omitted; refer to PDF] where the algebraic identity θ ( ^{q 2} ; ( ^{q 2 ) n ω 2} ) = ( ^{q 2 ) n (n +1 ) /2} ( ^{ω 2 ) n} θ ( ^{q 2} ; ^{ω 2} ) , that follows from ( 2.6), was used to obtain ( 6.2) and ( 6.4). Further details are presented in [ 21].

Proof of Proposition 4.1.

We first prove the estimate for the differences involving ^{Cos q} . From ( 6.1) and ( 6.2), one has [figure omitted; refer to PDF] where the triangle inequality gives the inequality in ( 6.6), and the evenness of | ^{ω 2} -q | / θ ( ^{q 2} ; ^{ω 2} ) allows for reduction to integration over [0 , ∞ ) in ( 6.6).

The change of variables ω =1 /u is made on the first integral in ( 6.7), and the algebraic identity θ ( ^{q 2} ; ^{u -2} ) = ^{u -2} θ ( ^{q 2} ; ^{u 2} ) is used to obtain [figure omitted; refer to PDF] Now ( 6.8) is used to reexpress the bound ( 6.7) as [figure omitted; refer to PDF]

From ( 2.9), we have that [figure omitted; refer to PDF] for 1 <q < ^{e π} . Deploying the bound ( 6.10) within the integral in the bound ( 6.9) gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It follows from [ 21] that [figure omitted; refer to PDF]

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).

Let α >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 , ^{e α ln (q )} ] and the second over [ ^{e α ln (q )} , ∞ ) . First, considering ω restricted to the interval [1 , ^{e α ln (q )} ] , one has that [figure omitted; refer to PDF] Now the function | ^{ω -1} -q ω | assumes its maximum value on [1 , ^{e α ln (q )} ] at the right endpoint ^{e α ln (q )} . In addition, for all q sufficiently close to 1 , the function | ω -q ^{ω -1} | assumes its maximum value on [1 , ^{e α ln (q )} ] at the right endpoint ^{e α ln (q )} . The latter statement follows since g ( ω ) = ω -q ^{ω -1} is increasing, and one needs only to compare endpoints |g (1 ) | =q -1 and |g ( ^{e α ln (q )} ) | = ^{e α ln (q )} -q ^{e - α ln (q )} to determine the larger value. An application of L'Hopital's rule gives that _{lim q [arrow right] ^{1 +}} |g ( ^{e α ln (q )} ) | / |g (1 ) | = ∞ . Thus, for q near 1 , on the interval [1 , ^{e α ln (q )} ] , one has [figure omitted; refer to PDF] Thus, we bound the integral in ( 6.14) by the length of the interval times the bound ( 6.15) on the numerator. This gives [figure omitted; refer to PDF] An application of L'Hopital's rule gives that [figure omitted; refer to PDF] which implies that in ( 6.16) we have [figure omitted; refer to PDF] Combining ( 6.18) with ( 6.16) and ( 6.14) gives that [figure omitted; refer to PDF]

We next estimate the portion of the integral in ( 6.11) over the interval [ ^{e α ln (q )} , ∞ ) . [figure omitted; refer to PDF] by bounding the integrals of the summands in ( 6.20). For each exponent k ∈ ... one has [figure omitted; refer to PDF] where a completion of squares gives ( 6.21). We use the following estimate from [ 21] [figure omitted; refer to PDF] which is applied to ( 6.21) with A =1 /ln (q ) ,B = -kln (q ) /2 , and C = ^{e α ln (q )} to obtain [figure omitted; refer to PDF] Combining ( 6.21) and ( 6.23) gives that, for each k ∈ ... , [figure omitted; refer to PDF] Revisiting ( 6.20) and applying ( 6.24) now gives [figure omitted; refer to PDF] Since the expression in ( 6.26) approaches 2 π ^{e - α 2} as q [arrow right] ^{1 +} , by choosing α sufficiently large one can make ( 6.26), and hence ( 6.25), arbitrarily small for q sufficiently close to 1 .

Now, let ... >0 be given. By ( 6.11) we have for all α >0 [figure omitted; refer to PDF] One has from ( 6.13) [figure omitted; refer to PDF] Choose A ~ >2 and then fix α >0 such that A ~ ^{e - α 2} < ... /2 . By ( 6.26) and ( 6.28), there exists _{q 1} = _{q 1} ( A ~ , α , ... ) such that for all 1 <q < _{q 1} one has [figure omitted; refer to PDF] For this value of α , by virtue of ( 6.19), one has [figure omitted; refer to PDF] which in turn says that there is a _{q 2} = _{q 2} ( α , ... ) such that for all 1 <q < _{q 2} one has [figure omitted; refer to PDF] Thus, for ^{Q C ...} =Q ( A ~ , α , ... ) =min { _{q 1} , _{q 2} } , and for all 1 <q < ^{Q C ...} , applying ( 6.31) and ( 6.29) to ( 6.27) yields [figure omitted; refer to PDF] independently of t ∈ ... . The proposition is now proven for ^{Cos q} .

The proof for ^{Sin q} parallels the above argument for ^{Cos q} , 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] with F (q ) defined in ( 6.12). For the integral in ω in ( 6.35) over the interval [ ^{e α ln (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 - α 2} still intact and the remaining factor approaching a constant as q [arrow right] ^{1 +} . On the other hand, the integral over the interval [1 , ^{e α ln (q )} ] in ( 6.35) proceeds as follows. One has [figure omitted; refer to PDF] where ( 6.36) and ( 6.37) hold for q sufficiently near ^{1 +} . Since _{lim x [arrow right]0} [ ( - ^{e -2 αx} + ^{e 2 αx} ) /x ] =4 α , we have the limit as q [arrow right] ^{1 +} of expression ( 6.37) is 0 . The remainder of the proof is now entirely parallel to that of the ^{Cos q} case.

Acknowledgments

The rogue data from the Draupner platform was graciously supplied by Dr. Sverre Haver from Statoil, Norway. The tsunami data came from DART buoys and was graciously provided by Dr. George Mongov at NOAA. The DART data is kept at the National Geophysical Data Center/World Data Center (NGDC/WDC) Historical Tsunami Database, Boulder, CO, USA. Figure 1profile was permitted for our use here by Dr. Benny Grosen. The authors are grateful for their assistance. Finally, they thank the referee for suggestions for improving the paper.

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