[show abstract][hide abstract] ABSTRACT: Two-dimensional (2D) interactions of two interfacial solitons in a two-layer fluid of finite depth are investigated under the assumption of a small but finite amplitude. When the angle between the wave normals of two solitons is not small, it is shown by a perturbation method that in the lowest order of approximation the solution is a superposition of two intermediate long wave (ILW) solitons and in the next order of approximation the effect of the interaction appears as position phase shifts and as an increase in amplitude at the interaction center of two solitons. When is small, it is shown that the interaction is described approximately by a nonlinear integro-partial differential equation that we call the two-dimensional ILW (2DILW) equation. By solving it numerically for a V-shaped initial wave that is an appropriate initial value for the oblique reflection of a soliton due to a rigid wall, it is shown that for a relatively large angle of incidence i the reflection is regular, but for a relatively small i the reflection is not regular and a new wave called stem is generated. The results are also compared with those of the Kadomtsev–Petviashvili (KP) equation and of the two-dimensional Benjamin–Ono (2DBO) equation.
Fluid Dynamics Research 10/2010; 42(6):065506. · 0.76 Impact Factor
[show abstract][hide abstract] ABSTRACT: We study the maximum wave amplitude produced by line-soliton interactions of the Kadomtsev-Petviashvili II (KPII) equation, and we discuss a mechanism of generation of large amplitude shallow water waves by multi-soliton interactions of KPII. We also describe a method to predict the possible maximum wave amplitude from asymptotic data. Finally, we report on numerical simulations of multi-soliton complexes of the KPII equation which verify the robustness of all types of soliton interactions and web-like structure. Comment: 28 pages, 11 figures. Stud. Appl. Math. vol. 122 (2009). Editorial production errors in the printed version were corrected. Several discussions about amplitude were improved. Some figures were improved
[show abstract][hide abstract] ABSTRACT: We consider the initial value problems of the Kadomtsev-Petviashvili (KP) equation for symmetric V-shape initial waves consisting of two semi-infinite line solitons with the same amplitude. Those are particularly important for studies of large amplitude waves such as tsunami in shallow water. Numerical simulations show that the solutions of the initial value problem approach asymptotically to certain exact solutions of the KP equation found recently in . We then use a chord diagram to explain the asymptotic result. This provides an analytical method to study asymptotic behavior for the initial value problem of the KP equation. We also demonstrate a real experiment of shallow water waves which may represent the solution discussed in this communication.
Journal of Physics A Mathematical and Theoretical 01/2009; 42(31). · 1.77 Impact Factor
[show abstract][hide abstract] ABSTRACT: An integrable two-component analogue of the two-dimensional long wave-short
wave resonance interaction (2c-2d-LSRI) system is studied. Wronskian solutions
of 2c-2d-LSRI system are presented. A reduced case, which describes resonant
interaction between an interfacial wave and two surface wave packets in a two
layer fluid, is also discussed.
[show abstract][hide abstract] ABSTRACT: We consider a mechanism of generation of huge waves by multi-soliton resonant interactions. A non-stationary wave amplification phenomenon is found in some exact solutions of the Kadomtsev-Petviashvili (KP) equation. The mechanism proposed here explains the character of extreme waves and of those in Tsunami.
[show abstract][hide abstract] ABSTRACT: The two-component analogue of two-dimensional long wave-short wave resonance interaction equations is derived in a physical setting. Wronskian solutions of the integrable two-component analogue of two-dimensional long wave-short wave resonance interaction equations are presented. Comment: 16 pages, 9 figures, revised version; The pdf file including all figures: http://www.math.utpa.edu/kmaruno/yajima.pdf
Journal of Physics A Mathematical and Theoretical 02/2007; · 1.77 Impact Factor
[show abstract][hide abstract] ABSTRACT: Oblique interaction of two solitons of the same amplitude in an extended Kadomtsev-Petviashvili (EKP) equation, which is a weakly two-dimensional generalization of an extended Korteweg-de Vries (EKdV) equation, is investigated. This interaction problem is solved numerically under the initial and boundary condition simulating the reflection problem of the obliquely incident soliton due to a rigid wall. The essential parameters are given by Q*\equiv aQ and Omega*\equivOmega/a1/2. Here, Q is the coefficient of the cubic nonlinear term in the EKP quation, a the amplitude of the incident soliton and Omega\equiv\tanthetai, thetai being the angle of incidence. The numerical solutions for various values of these parameters reveal the effect of the cubic nonlinear term on the behavior of the waves generated by the interaction. When Q* is small, the interaction property is very similar to that of the Kadomtsev-Petviashvili equation. Especially, for relatively small Omega*, a new wave of large amplitude and of soliton profile called ``stem'' is generated. On the other hand, when Q* is close to 6, no stem is generated owing to the existence of amplitude restriction for the soliton solution.
Journal of the Physical Society of Japan 01/2007; 76(8). · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: Studies on the oblique interactions of weakly nonlinear long waves in dispersive systems are surveyed. We focus mainly our concentration on the two-dimensional interaction between solitary waves. Two-dimensional Benjamin–Ono (2DBO) equation, modified Kadomtsev–Petviashvili (MKP) equation and extended Kadomtsev–Petviashvili (EKP) equation as well as the Kadomtsev–Petviashvili (KP) equation are treated. It turns out that a large-amplitude wave can be generated due to the oblique interaction of two identical solitary waves in the 2DBO and the MKP equations as well as in the KP-II equation. Recent studies on exact solutions of the KP equation are also surveyed briefly.
Fluid Dynamics Research 01/2006; · 0.76 Impact Factor
[show abstract][hide abstract] ABSTRACT: A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.
Journal of the Physical Society of Japan 01/2004; 73(4):831-837. · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: Two-dimensional interaction of two solitary waves is investigated numerically on the basis of a Modified Kadomtsev-Petviashvili equation. Two types of interaction are dealt with; one is the interaction of two positive solitary waves of an equal amplitude and the other is that between positive and negative solitary waves of an equal amplitude. The latter turns out to be considerably different from the former. In both cases the characteristics of the interaction depend on the ratio of a parameter representing the difference of the propagation directions of two solitary waves to the amplitude of the solitary waves. It is found that a new wave of a fairly large amplitude is produced for some range of the ratio as a result of the interaction of two positive solitary waves.
Journal of the Physical Society of Japan 01/2004; 73(11):3034-3043. · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: Discretization of the Bruschi-Ragnisco lattice is investegated by singularity confinement test. The equation is linearized by the Cole-Hopf like transformation.
[show abstract][hide abstract] ABSTRACT: Oblique interaction of internal solitary waves in a two-layer fluid system with infinite depth is studied. Two-dimensional Benjamin–Ono (BO) equation is solved numerically to investigate the strong interactions of the non-linear long waves whose propagation directions are very close to each other. Computations of time development are performed for two initial settings: the first one is superposition of two BO solitons with the same amplitude and with different propagation directions, and the second one is an oblique reflection of a BO soliton at a vertical wall. It is observed that the Mach reflection does occur for small incident angles and for some incident angles very large stem waves are generated.
Fluid Dynamics Research 01/2001; · 0.76 Impact Factor
[show abstract][hide abstract] ABSTRACT: Three-dimensional instabilities oftwo-dimensional periodic capillary gravity waves of permanent formnear the fourth harmonic resonance are investigated numerically.It is confirmed that the unstable regions appearin the neighborhood of the linear resonance curves of sum interactionsassociated with a fundamental mode and its fourth harmonic.No unstable regions overlap in the wave number plane of disturbances.Two regions are frequently reconnected.
Journal of The Physical Society of Japan - J PHYS SOC JPN. 01/1997; 66(9):2675-2681.
[show abstract][hide abstract] ABSTRACT: Three systems of weakly nonlinear envelope equations for a fundamental mode and its fourth harmonic of two-dimensional capillary gravity waves on deep water are derived.These systems are used to investigate stability of progressive periodic waves of permanent form.It is found that the permanent waves in which the fundamental mode and its fourth harmonic are of the same order are unstable.Modulational instability in which the dominant mode in the eigenfunction is near the fourth harmonic is found in the third order approximation.Furthermore, instability related to the fourth harmonic resonance is found in the fourth order approximation.
Journal of the Physical Society of Japan 01/1997; 66(9):2665-2674. · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: Fully nonlinear time evolution of periodic capillary gravity waveson an inviscid fluid of infinite depth is investigated numerically.The cases where the effect of surface tension is weaker than that of gravityare considered.It is found that the weakly resonant symmetric steady waves are unstableto two kinds of disturbances: the sidebands and the higher harmonics.Ripples are generated within a time of a few periods.They first appear at the point with the greatest slope on the forward face of the highest crest of the disturbed steady waves.
Journal of the Physical Society of Japan 01/1996; 65(7):2060-2067. · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: The time evolution of a nonlinear modulated wavetrain is investigated using the Dysthe equation with an additional nonlocal nonlinear damping term. It is found that the additional damping term has no significant effect on qualitative features of the modulational instability of the Stokes wave. Numerical results show that for appropriate values of the coefficient of the damping term, it is practically effective only near the crest of envelope, and causes the wave number downshift. These results suggested that two factors are essential to cause the downshift. The first is the enough nonlinearity to produce the asymmetry in spectral distribution. The second is the nonlinear dissipation that affects especially the higher components only when the wavetrain is strongly modulated.
Journal of the Physical Society of Japan 01/1995; 64(12):4660-4669. · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: Two-dimensional interaction of internal solitary waves is studied for a two-layer fluid with a thickness ratio close to a certain critical one depending on the density ratio. It is described by a Modified Kadomtsev-Petviashvili (MKP) equation, provided the propagation directions of the waves are close to each other. The MKP equation is investigated numerically and it is found that the interaction almost satisfying the condition of soliton resonance occurs when the wave amplitudes are small. The interaction is qualitatively similar to the soliton resonance in the Kadomtsev-Petviashvili (KP) equation except that a deformation of the newly generated waves and a radiation can be seen.
Journal of the Physical Society of Japan 01/1993; 62(11):3881-3892. · 2.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: The linear stability of two-dimensional steady flows between two long, eccentric, rotating circular cylinders is studied numerically under the condition that the inner cylinder rotates uniformly while the outer one is at rest. By using the pseudospectral method it is found that the critical Reynolds number increases with the eccentricity \varepsilon. The critical axial wave number is found to remain nearly constant for small \varepsilon and to increase with larger \varepsilon. The eigenfunctions are distributed in the region from the position of the maximum gap to 180° downstream of that position. The Taylor-vortexlike three-dimensional steady flows are computed for several supercritical Reynolds numbers. The torques acting on the cylinders and the position of maximum vortex activity are calculated.
Journal of The Physical Society of Japan - J PHYS SOC JPN. 01/1989; 58(7):2355-2364.