dV Equation Research Articles

Exponentially small estimate for a generalized solitary wave solution to the perturbed K−dV equation

Exponentially small estimate for a generalized solitary wave solution to the perturbed K−dV equation

Hydrodynamic collapse of a cylindrical cavity on dry sea bed

A theory is developed on the generation and collapse of water craters created by explosions in shallow water. The crater lip generates a leading wave which radiates outward while the water cradle collapses towards the center. This collapse is similar to the dam break problem on a dry bed except for the cylindrical symmetry and is solved using the nonlinear long wave equations and the method of characteristics. When the water edge reaches the center, it rebounds on itself vertically and forms a cylindrical dissipative bore radiating outward. The bore height decays rapidly with distance from the center and is transformed into a nonlinear undular bore obeying the K dV equation. The K dV equation is extended to the Stokesian range for the higher frequency waves that follow. The theory is applicable in the reverse direction, i.e. for negative time steps. Therefore, given an experimental wave record obtained at a fixed distance from the center, it allows us to determine the size of the crater which would generate such a record. The method is applied to wave records obtained in the laboratory, ponds, and lakes with TNT explosions and also tested with the wave records from nuclear explosions, with yields ranging over 11 orders of magnitude. It is found consistently that the physical water crater created by underwater explosions in shallow water follows, R c = 4.4 · W 0.25, R c , in feet, and W is the explosion yield in lb of TNT. The relative hydrodynamic energy dissipation is nearly a constant 40% of the potential energy of the initial crater with lip.

Floquet exponents for Jacobi fields

AbstractThis paper introduces a Riemannian invariant of a compact Riemannian manifold based on the spectral theory for the Jacobi field operator. It is the Floquet exponent for this operator, a purely dynamical quantity computable directly from the asymptotic behavior of Jacobi fields. We show that it is related to certain traces of the Green's function, and we derive further regularity and analyticity properties for the Green's function. In case the geodesic flow is ergodic, the Floquet exponent generalizes the measure entropy, and several entropy estimates follow. An asymptotic expansion of the Floquet exponent gives rise to a sequence of ‘Jacobi invariants’, which are related to the polynomial invariants of the K dV equation.

Experimental Srudy of an Ion-Acoustic Soliton with Dissipation

The propagation of an ion-acoustic soliton in an Ar (or Ar–He) plasma is investigated experimentally in a double plasma device. The various patterns of signals propagating from the separation grid into the target region are classified into three types, Besides the normal soliton pattern, there appear signals with either a shallow dip or a deep valley. The K–dV equation with a term expressing the effect of ion Landau damping is solved numerically with parameters corresponding to the experimental conditions. One of the types observed in the experiment (with a shallow dip) is fully reproduced theoretically with the corresponding plasma parameters. Effects of the initial height of the perturbation on the evolution of the signal are investigated experimentally and theoretically. The minority He ion which causes an increase of the ion Landau damping rate had the same effect as the decrease of the initial height of the signal.

Viscous Effects Can Destabilize Linear and Nonlinear Water Waves

Long waves on a running stream in shallow water are shown theoretically to be susceptible, in some circumstances, to a viscous instability, which can lead to rapid linear and nonlinear growth. The theory is based on high Reynolds numbers and involves viscous-inviscid interplay, leading in effect to a viscosity-modified version of the classical nonlinear K dV equation. This is with a pre-existing mean flow present. The modification is due to a Stokes wall layer and it can cause severe linear and nonlinear instability. A model profile for the original mean flow is studied first, followed by a smooth realistic profile, the latter provoking a nonlinear critical layer in addition. The theory is linked with interactive-boundary-layer analysis and linear and nonlinear Tollmien-Schlichting waves and there is some analogy with the recent findings (in work by the authors) of nonlinear break-ups occurring in any unsteady interactive boundary layer, including the external boundary layer and internal channel or pipe flows.

On the Bäcklund transformation and Hamiltonian properties of superevaluation equations

Bäcklund and Darboux–Bäcklund transformations are deduced for the superevaluation equations recently deduced by Kupershmidt [B. Kuperschmidt, ‘‘A super-K–dV equation—An integrable system,’’ preprint UTSI-Tullahoma, 1984; Phys. Lett. A 102, 213 (1983); J. Phys. A 17, L 863 (1984)] and Gürses [H. Gürses and O. Oguz, Phys. Lett. A 108, 437 (1985)]. By a extension of the technique of BPT [M. Boiti, F. Pempinelli, and G. Z. Tu, Nuovo Cimento B 79, 231 (1984)] to anticommuting variables the bi-Hamiltonian structure and hence the form of the recursion operator for the Lie–Bäcklund symmetry for such equations are deduced. Incidentally some explicit forms of the Lie–Bäcklund symmetry are also deduced.

Excitation of ion-acoustic rarefactive solitons in a two-electron-temperature plasma

Rarefactive ion-acoustic solitons have been observed in a two-electron-temperature plasma. Some of the characteristics can be interpreted by the solution of the Kortweg–de Vries (K–dV) equation. The Mach number of the solitons is a function of the temperature ratio of hot and cold components.

Experimental studies of potential problems in quantum mechanics using nonlinear transmission line

The experimental method for solving the potential problem in quantum mechanics is demonstrated using the nonlinear transmission line equivalent to the Korteweg–de Vries equation (K–dV equation) Ut−6UUx+Uxxx=0. An input voltage pulse corresponding to the potential is observed to dissolve into a finite train of solitons propagating along the line. The signal voltage consisting of solitons is compared with the solution of the initial-value problem for the K–dV equation, which is exactly solved by the theory called the ‘‘inverse scattering method.’’ Consequently, it is shown that the number of bound states and discrete energy levels for one-dimensional potentials (square well potential, −U0 sech2 αx potential, and the Kronig–Penny potential) are determined by observing the signal waveform without calculations.

Ion-acoustic solitary waves in relativistic plasmas

This is a sequel to our earlier study on ion-acoustic waves studied through the augmentation to a modified Korteweg–deVries (K–dV) equation. We have derived a K–dV equation in a plasma, taking account of weakly relativistic effects, and the result shows that the solitary wave does exhibit the relativistic effect in the presence of ion streaming.

An inverse problem for a differential operator with a mixed spectrum

and that the continuous spectrum of L consists of only a finite number of finite intervals and one infinite interval. Under these assumptions it will be shown that the specification of a finite number of eigenvalues will lead to the determination and construction of a unique potential q(t). The authors plan to discuss the relationship between these potentials and the K dV equation in a future publication.

On the establishment of oscillatory acoustic shock waves in monorelaxing fluids

Analytical approximations and numerical solutions exhibiting oscillatory shock formation in mono‐relaxing fluids have been obtained for plane and spherical finite‐amplitude waves. By comparing these results with previously reported oscillatory solutions of the Korteweg—de Vries (K—dV) and K—dV—Burgers' equation for plane wave propagation in a purely dispersive medium, necessary conditions have been established for the formation of osciliatory shock waves in mono‐relaxing fluids. These conditions which are expressed in the form of dimensionless parameters of the medium and the propagating wave have been evaluated for particular relaxing fluid media. Several examples of the radiated intensities required to realize oscillatory acoustic shock waves at selected frequencies in such media will be provided. The “solitonlike” property of these oscillatory shock waves and those of the ideal “soliton” solutions of the longless K—dV equation will then be considered.

A New Approach for Deriving the K-dV Equation

A New Approach for Deriving the <i>K</i>-dV Equation