Invariants of the two dimensional Korteweg-de vries and Kadomtsev-Petviashvili equations

Eleven invariants of the two dimensional Korteweg-de Vries and Kadomtsev-Petviashvili equations are found by a new method. They include the six invariants that can be obtained from Lie group theory. The new method also reproduces all Lie invariants of the Maxwell and nonlinear Schrödinger equations.     

Pseudopotentials and the inverse scattering problems of the generalized Korteweg-de vries equation

The algebraic prolongation structure approach of Wahlquist and Estabrook is used to determine the various forms of inverse scattering equations known for the generalized Korteweg-de Vries equation.     

A comparison between the cnoidal wave and an approximate periodic solution to the Korteweg-de Vries equation

Summary The motion of small-but finite-amplitude waves in shallow water is often modeled by the well-known Korteweg-de Vries (KdV) equation. Here we consider a case in which no solitons are present and compare the exact periodic travelling-wave solutions of the KdV equation (the cnoidal wave) to an approximate periodic solution of this equation previously obtained by the authors. We find that the approximate wave form is graphically indistinguishable from the cnoidal wave for a wide range of wave amplitudes. Furthermore, by extending the amplitude range up to the breaking wave limit we find that the approximate wave form is still a close representation of the cnoidal wave. This suggests that the approximate solution, which is just a simple formula, might be used for many practical calculations in place of the more difficult to compute cnoidal wave.

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Riassunto L'evoluzione in acqua bassa di onde di ampiezza piccola ma finita è spesso descritta dalla classica equazione di Korteweg-de Vries (KdV). è qui considerato un caso senza solitoni e l'esatta soluzione stazionaria periodica dell'equazione di KdV (nota come onda cnoidale) è confrontata con una soluzione periodica approssimata precedentemente ottenuta dagli autori. I risultati del confronto mostrano che la soluzione approssimata è graficamente indistinguibile dall'onda cnoidale in un ampio intervallo di ampiezze d'onda. Aumentando l'ampiezza d'onda fino al limite della rottura dell'onda stessa si mostra che la soluzione approssimata è ancora una buona rappresentazione dell'onda cnoidale. Questi risultati suggeriscono che la soluzione approssimata, espressa da una formula molto semplice, possa essere usata in molti calcoli applicativi al posto della piú complicata onda cnoidale.

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Резюме Движение волн с малой, но конечной амплитудой на мелкой воде часто моделируется с помощью хорошо известного уравнения Кортевега-де Вриса. В этой работе мы рассматриваем случай отсутствия солитонов и сравниваем точное периодическое распространяющее волновое решение уравнения Кортевега-де Вриса (?кноидальная? волна) с приближенным периодическим решением этого уравнения, ранее полученным авторами. Мы находим, что приближенная волновая форма графически наразличима от ?кноидальной? волны в широкой области амплитуд. Кроме того, распространяя область амплитуд вплоть до нарушення волнового предела, мы получаем, что приближенная волновая форма еще хорошо аппроксимирует ?кноидальную? волну. Этот результат позволяет предположить, что приближенное решение, которое описывается простой формулой, мож⪟т быть использовано в большом числе практических вычислений вместо более сложного выражения.

     

Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations

In this paper we establish new results about the existence, stability, and instability of periodic travelling wave solutions related to the critical Korteweg-de Vries equation

u_t+5u⁴u_x+u_xxx=0,     

Breather-like solution of the Korteweg-de Vries equation

A new real singular solution of the Korteweg-de Vries equation is briefly described. It is shown that the spectrum of the associated linear problem consists of a pair of complex conjugate eigenvalues. The existence of constants of motion and eigenfunction normalization integral is also discussed.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

Group-theoretical interpretation of the Korteweg-de Vries type equations

The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schrödinger equation (with nonlocal potential) plays the same role as the one-dimensional Schrödinger equation does in the theory of the Korteweg-de Vries equation.     

A direct method to find solutions of some type of coupled Korteweg-de Vries equations using hyperelliptic functions of genus two

We suggest how one can obtain exact solutions of some type of coupled Korteweg-de Vries equations by means of hyperelliptic functions of genus two.     

Korteweg-de Vries hierarchy using the method of base equations

A base-equation method is implemented to realize the hereditary algebra of the Korteweg-de Vries (KdV) hierarchy and the N-soliton manifold is reconstructed. The novelty of our approach is that, it can in a rather natural way, predict other nonlinear evolution equations which admit local conservation laws. Significantly enough, base functions themselves are found to provide a basis to regard the KdV-like equations as higher order degenerate bi-Lagrangian systems.     

Exact periodic solution in coupled nonlinear Schodinger equations

The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.     

The Korteweg-de Vries hierarchy: structure and solution

We attempt to realize the structure of the Korteweg-de Vries (KdV) hierarchy by using a simple dimensional analysis. The specific results presented refer to equations of the hierarchy and conserved Hamiltonian densities associated with them. Based on the Gel’fand-Levitan-Marchenko equation we construct a series expansion for the unstable solution of the KdV-like equations by using the continuous part of the spectrum for the Schrödinger operator. Our result is in exact agreement with that obtained from the Sabatier’s formulation of the inverse problem for rational reflection coefficients.     

A simple solution to the strongCP-problem

We suggest soft breaking of Peccei-Quinn symmetry as a simple solution to the strongCP problem without the presence of the axion. In the context of theSU(2) _L ×U(1) model, it is shown how settingθ _tree=0 keeps θ calculable and, furthermore, extremely small: θ?10^?16. Unlike in the case of the axion the model is free from the cosmological domain wall problem. Possible extension of this idea to grand unification is discussed.     

An exact solution of the Korteweg-de Vries equation with dissipation

By means of the method proposed in the papers [1, 2] we look for solutions of the Korteweg—de Vries equation with dissipation. A new solution is found and expressed by means of the Weierstrass P-function.Partially supported by NSF Grant No. INT 73.20002 A01 formerly GF-41958.     

The integration of Burgers and Korteweg-de Vries equations with nonuniformities

In this letter we demonstrate that both Burgers and Korteweg-de Vries equations with nonuniformity terms can be reduced to a Burgers or Korteweg-de Vries equation with constant coefficients if these terms satisfy a compatibility condition.     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

Evolution of radiating solitons described by the fifth-order Korteweg-de Vries type equations

The time dependence of velocities and amplitudes of radiating solitons, described by the fifth-order Korteweg-de Vries type equations is investigated.