Dispersive Water Wave Equation Research Articles

Non-isospectral scattering problems and truncation for hierarchies: Burgers and dispersive water waves

Non-isospectral scattering problems have been proven useful for several reasons, amongst them the information that they provide about Painlevé truncation for entire hierarchies of integrable partial differential equations (PDEs). We show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1 + 1 dimensions, namely Burgers' hierarchy and the dispersive water wave hierarchy. Burgers' equation is well-known as a model for turbulence, and the dispersive water wave equations as a system governing shallow water waves. We also see how these results are easily extended to related hierarchies in 2 + 1 dimensions. Our results for the 2+1-dimensional Burgers' hierarchy, and for the 1 + 1 and 2+1-dimensional dispersive water wave hierarchies, are all new.

Projective structure and integrable geodesic flows on the extension of Bott‐Virasoro group

This is a sequel to our paper (Lett. Math. Phys. (2000)), triggered from a question posed by Marcel, Ovsienko, and Roger in their paper (1997). In this paper, we show that the multicomponent (or vector) Ito equation, modified dispersive water wave equation, and modified dispersionless long wave equation are the geodesic flows with respect to an L2 metric on the semidirect product space , where Diffs(S1) is the group of orientation preserving Sobolev Hs diffeomorphisms of the circle. We also study the projective structure associated with the matrix Sturm‐Liouville operators on the circle.

COMPUTERIZED SYMBOLIC COMPUTATION FOR THE MODIFIED DISPERSIVE WATER WAVE EQUATIONS

With computerized symbolic computation, we present an auto-Bäcklund transformation and three families of soliton-like solutions to the modified dispersive water wave equations.

Integrable Geodesic Flows on the (Super)extension of the Bott–Virasoro Group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys.40 (1997), 31–39). The Ito equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L2 metric on the semidirect product space Diff s \(({\text{S}}^{\text{1}} \widehat{) \odot }\)C∞(S1), where Diff s (S1) is the group of orientation-preserving Sobolev Hs diffeomorphisms of the circle. We also study the geodesic flows with respect to H1 metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu–Schwarz space.

Nonhomogeneous hydrodynamic-type equations and their superextension

It is observed that some (x,t) interchanged nonlinear integrable system do have the characteristics of nonhomoneous integrable system of hydrodynamic type. The second Hamiltonian structure obtained through the Miura map or Lie derivative with respect to the master symmetry do have the requisite features. In the present communication we study the cases of dispersive water wave and NLS equation with (x,t) interchanged. Along with these we also analyse the case of supersymmetric version of the (x,t) interchanged KdV problem which yields an example of a super hydrodynamics system.

A Method for Obtaining Darboux Transformations

In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation.

A time–dependent nonlinear mild slope equation for water waves

A weakly nonlinear and dispersive water wave equation, which in linearized form yields a new version of the time–dependent mild–slope equation of Smith and Sprinks (1975), is derived. The applicable spectral width of the new wave equation for random waves is found to be more satisfactory than that of Smith and Sprinks (1975). For very shallow depths the equation reduces to the combined form of Airy9s nonlinear non–dispersive wave equations; if the lowest–order dispersion is retained it produces the combined form of Boussinesq9s equations. In the deep–water limit the equation admits the second–order Stokes waves as analytical solutions. Furthermore, by introducing a right–moving coordinate transformation, the equation is recast into a unidirectional form, rendering the KdV equation in one limit while reproducing the second–order Stokes waves in the other.

A supersymmetric dispersive water wave equation

A supersymmetric extension of the classical dispersive water wave equation is proposed. The system is shown to be tri-Hamiltonian. In particular, the second Hamiltonian structure is analogous to the N = 2 superconformal algebra. A gauge map to the even-parity super KP system is exhibited.

Symmetry Reductions, Exact Solutions, and Painlevé Analysis for a Generalized Boussinesq Equation

In this paper, new nonclassical symmetry reductions and exact solutions are presented fora Generalised Boussinesq equation u xxxx + pu tu xx + qu xu xt + ru 2 xu xx + u tt =O, which has the modified Boussinesq equation ( q = O, r = − 1 2 p 2) and dispersive water wave equation, or classical Boussinesq equations ( q = 2 p, r = 3 2 p 2) as special cases. These symmetry reductions are obtained using the Direct Method, originally developed by Clarkson and Kruskal to study symmetry reductions of the Boussinesq equation, which involves no group theoretic techniques, and using these reductions, we obtain exact solutions expressible in terms of solutions of the second and fourth Painlevé equations, Jacobi and Weierstrass elliptic functions, and elementary functions, for certain values of the parameters p, q, and r. Furthermore, in the case when q = p and r = 1 2 p 2, symmetry reductions are obtained which are reminiscent of reductions of the 2 + 1-dimensional cubic nonlinear Schrödinger equation arising from the Talanov lens transformation.

Infinite Dimensional Lie Algebra and Symmetries of Nonlinear Equations in Three Dimension — An Overview

Symmetries of nonlinear systems integrable via inverse scattering transform in three dimension are analysed from a different point of view. Actually the classical approach of Lie Backlund transform is not so effective due to the complexity of the computations involved. So we review here the new approach for finding symmetries on the basis of Lax algebra. In the first part we mention in short the earlier techniques for the study of symmetries and in the second part the present method is reviewed in detail following the earlier work of Li Yi Shen et al., with an eye for application to nonlinear systems in three dimension. The approach is highly effective in finding symmetries of non-local, nonlinear in equations in three dimension, which is otherwise impossible to find out. The methodology is illustrated with the help of three examples. (1) the Harry Dym equation in three dimension, (2) the non-local KP equation and (3) dispersive water wave equation in three dimension. Lastly we demonstrate that these symmetries generate an infinite dimensional Lie algebra through a suitably defined commutator.

Dispersive water-wave equations: a paradigm of the Painlev$eacute$ conjecture

Dispersive water-wave equations: a paradigm of the Painlev$eacute$ conjecture

Dispersive water-wave equations: a paradigm of the Painleve conjecture

The authors have analysed the equation of dispersive water waves within the formalism of a singular point analysis of the Painleve conjecture. It is interesting to observe that the two different similarity reductions lead to apparently different-looking ordinary differential equations with a similar singularity structure. While one is a pure Painleve equation, the other is not. On the other hand, a direct analysis of the PDE itself shows resonances at r=-1, 2, 3, 4 which contain those of the ODE. If, however, they adopt the exact procedure of Weiss et al. (1983) and try to truncate the expansion then it is seen that the set of equations obtained are not overdetermined. Thus it is not possible to derive the Lax pair via the Painleve analysis but a Backlund transformation can be written. The number of arbitrary coefficients is seen to be in conformity with the Cauchy-Kowalevskaya theorem.

Vector extensions of modified water wave equations

An integrable multicomponent extension is found of the modified Korteweg-de Vries (mKdV) and modified dispersive water wave (mDWW) equations. The former is realized as a factor system of the latter.

A multicomponent water wave equation

A vector extension of the classical dispersive water wave equation is shown to be a member of an infinite integrable three-Hamiltonian hierarchy.

Mathematics of dispersive water waves

A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.