Solutions associated with discrete and continuous spectrums in the inverse scattering method for the Vakhnenko-Parkes equation

In this paper the inverse scattering method is applied to the Vakhnenko-Parkes equation. We describe a procedure for using the inverse scattering transform to find the solutions that are associated with both the bound state spectrum and continuous spectrum of the spectral problem. The suggested special form of the singularity function gives rise to the multi-mode periodic solutions. Sufficient conditions are obtained in order that the solutions become real functions. The interaction of the solitons and multi-mode periodic waves is studied. The procedure is illustrated by considering a number of examples

Special singularity function for continuous part of the spectral data in the associated eigenvalue problem for nonlinear equations

The procedure for finding the solutions of the Vakhnenko-Parkes equation by means of the inverse scattering method is described. The continuous spectrum is taken into account in the associated eigenvalue problem. The suggested special form of the singularity function for continuous part of the spectral data gives rise to the multimode solutions. The sufficient conditions are proved in order that these solutions become real functions. The interaction of the N periodic waves is studied. The procedure is illustrated by considering a number of example

A novel nonlinear evolution equation integrable by the inverse scattering method

A Backlund transformation for an evolution equation (ut+u ux)x+u=0 transformed into new coordinates is derived. An inverse scattering problem is formulated. The inverse scattering method has a third order eigenvalue problem. A procedure for finding the exact N-soliton solution of the Vakhnenko equation via the inverse scattering method is described

The inverse problem for some special spectral data

In this paper the spectral problem of third order for the inverse scattering transform (IST) method is solved. For the discrete part of the spectral data, the two-multiple poles are taken into account. The line spectrum of continuum states for the IST method is examined as well. The suggested spectrum approximates in first order the step-function. The scope for the sug- gested spectral data is demonstrated through the analysis of the Vakhnenko–Parkes equation that allows new solutions to be obtained. The account of the time-dependence is different from the standard procedure

Approach in theory of nonlinear evolution equations : the Vakhnenko-Parkes equation

A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. It is shown how the equation arises in modelling the propagation of high-frequency waves in a relaxing medium. The VE is related to a particular form of the Whitham equation. Periodic and solitary traveling wave solutions are found by direct integration. Some of these solutions are loop-like in nature. The VE can be written in an alternative form, now known as the Vakhnenko-Parkes equation (VPE), by a change of independent variables. The VPE can be written in Hirota bilinear form. It is then possible to show that the VPE satisfies the ‘N-soliton condition’, in other words that the equation has an N-soliton solution. This solution is found by using a blend of the Hirota method and ideas originally proposed by Moloney & Hodnett. This solution is discussed in detail, including the derivation of phase shifts due to interaction between solitons. Individual solitons are hump-like in nature. However, when transformed back into the original variables, the corresponding solution to the VE comprises N loop-like solitons. It is shown how aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, can be used to find one and two-soliton solutions to the VPE even though, in contrast to the KdV equation, the VPE’s spectral equation is not second-order (the isospectral Schr¨odinger equation). A B¨acklund transformation is found for the VPE and this is used to construct conservation laws. It is shown that the specral equation for the VPE is actually third-order. Then, based on ideas of Kaup and Caudrey, the standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to N-soliton solutions andM-mode periodic solutions respectively. Interactions between these types of solutions are investigated

Quasistatic loading of Berea sandstone

Запропонована феноменологiчна модель для опису властивостей напруження-деформацiя пiсковика пiд дiєю повiльного навантаження. Розглянута комбiнацiя трьох механiзмiв, якi пов’язуються з внутрiшнiми обмiнними процесами: механiзм стандартного релаксуючого твердого тiла, пружний механiзм з прилипанням, механiзм залишкової пластичної деформацiї. З малою кiлькiстю параметрiв модель вiдтворює як якiсно, так i кiлькiсно головнi експериментальнi данi по напруженню-деформацiї для пiсковика Береа. Модель правильно вiдтворює великi та малi петлi на траєкторiї напруження-деформацiя (пам’ять про кiнцеву точку). Власне запропонована залежнiсть деформацiї вiд напруження є не чим iншим, як рiвнянням стану пiсковика

The inverse scattering method for the equation describing high-frequency waves in a relaxing medium

This chapter looks at the inverse scattering method for the equation describing high-frequency waves in a relaxing mediu

The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method

A procedure for finding the solutions of the Vakhnenko–Parkes equation by means of the inverse scattering method is described. Both the bound state spectrum and the continuous spectrum are considered in the associated eigenvalue problem. The suggested special form of the singularity function gives rise to periodic solutions. The interaction of a soliton with a one-mode periodic wave is studied

Periodic and solitary-wave solutions of the Degasperis-Procesi equation

Travelling-wave solutions of the Degasperis-Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis-Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered

Explicit solutions of the Camassa-Holm equation

Explicit travelling-wave solutions of the Camassa-Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis-Procesi equation are given in an appendix