Q-advanced models for tsunami and rogue waves

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

Theta function identities in the study of wavelets satisfying advanced differential equations

AbstractThe study of wavelets that satisfy the advanced differential equation K′(t)=K(qt) is continued. The connections linking the theories of theta functions, wavelets, and advanced differential equations are further explored. A direct algebraic–analytic estimate is given for the maximal allowable translation parameter N(q) such that b<N(q) guarantees Λ(0,q,b)≡{(qm/2/c0)K(qmt−nb)|m,n∈Z} is a wavelet frame for L2(R), where c0 is the L2 norm of K. For any q>1 and any b>0 we find conditions guaranteeing that Λ(p,q,b)≡{(qm/2/‖K(p)‖)K(p)(qmt−nb)|m,n∈Z} is a wavelet frame for L2(R) where K(p) denotes the pth derivative/antiderivative of K. The frames Λ(p,q,b) become snug as either p→−∞ or q→∞, and their lower frame bounds A(p,q,b)→∞ as q→∞

Q-advanced models for tsunami and rogue waves

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 &gt;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