Conservation Laws,Hamiltonian Structure,Modulational Instability Properties and Solitary Wave Solutions for a Higher‐Order Model Describing Nonlinear Internal Waves

Recent theoretical advances in connecting the wave‐induced mean flow with the conserved pseudomomentum per unit mass has permitted the first rational derivation of a model that describes the weakly nonlinear propagation of internal gravity plane waves in a continuously stratified fluid. Depending on the particular parameter regime examined the new model corresponds to an extended bright or dark derivative nonlinear Schrödinger equation or an extended complex‐valued modified Korteweg‐de Vries or Sasa–Satsuma equation. Mass, momentum, and energy conservation laws are derived. A noncanonical infinite‐dimensional Hamiltonian formulation of the model is introduced. The modulational stability characteristics associated with the Stokes wave solution of the model are described. The bright and dark solitary wave solutions of the model are obtained.     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

The tanh method: a tool for solving certain classes of non‐linear PDEs

The tanh (or hyperbolic tangent) method is a powerful technique to look for travelling waves when dealing with one‐dimensional non‐linear wave and evolution equations. In particular, this method is well suited for those problems where dispersion, convection and reaction–diffusion play an important role. To show the strength of this method we study a coupled set (the so‐called Boussinesq equations) which arises in the theory of non‐linear dispersive water waves. As a result, a solitary wave profile is found which generalizes an earlier result, the famous Korteweg‐de Vries solitary wave solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Solitary waves in a fluid-filled thin elastic tube with variable cross-section

The present work treats the arteries as a thin walled prestressed elastic tube with variable cross-section and uses the longwave approximation to study the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg–de Vries equation with a variable coefficient. It is shown that this type of equations admits a solitary wave type of solution with variable wave speed. It is observed that, for soft biological tissues with an exponential strain energy function the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Perturbations of Soliton Solutions to the Unstable Nonlinear Schrödinger and Sine-Gordon Equations

The adiabatic evolution of soliton solutions to the unstable nonlinear Schrödinger (UNS) and sine-Gordon (SG) equations in the presence of small perturbations is reconsidered. The transport equations describing the evolution of the solitary wave parameters are determined by a direct multiple-scale asymptotic expansion and by phase-averaged conservation relations for an arbitrary perturbation. The evolution associated with a dissipative perturbation is explicitly determined and the first-order perturbation fields are also obtained.     

Bäcklund transformation,superposition formulae and N-soliton solutions for the perturbed Korteweg–de Vries equation

With symbolic computation, under investigation in this paper is the perturbed Korteweg–de Vries equation for the nonlocal solitary waves and arrays of wave crests. Via the Hirota method, the bilinear form, Bäcklund transformation and superposition formulae are obtained. N-soliton solutions in terms of the Wronskian are constructed. Asymptotic analysis is used to analyze the collision dynamics, and figures are plotted to illustrate the influence of the perturbation. We find that the perturbation affects the propagation velocities of the solitons, but does not affect the amplitudes and widths of the solitons. Besides, the solitonic collisions turn out to be elastic.     

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg‐de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh‐order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite‐order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite‐difference discretization, in which a lattice generalized solitary wave solution is found.     

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

In the small‐dispersion limit, solutions to the Korteweg—de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg—de Vries solution near the leading edge of the oscillatory zone up to second‐order corrections. This expansion involves the Hastings‐McLeod solution of the Painlevé II equation. We prove our results using the Riemann‐Hilbert approach. © 2009 Wiley Periodicals, Inc.     

Topological and non-topological exact soliton solution of the power law KdV equation

This paper obtains the exact 1-soliton solution of the perturbed Korteweg–de Vries equation with power law nonlinearity. Both topological as well as non-topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. Finally, the numerical simulations are implemented in the paper.     

Adiabatic parameter dynamics of perturbed solitary waves

The soliton perturbation theory is used to study the solitons that are governed by the generalized Korteweg–de Vries equation in the presence of perturbation terms. The adiabatic parameter dynamics of the solitons in the presence of the perturbation terms are obtained.     

Numerical algorithms for solutions of Korteweg–de Vries equation

The nonlinear Korteweg–de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion‐based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Slowly Varying Solitary Wave Solutions of the Perturbed Korteweg-de Vries Equation Revisited

Using a multi-scale perturbation expansion we reconsider the slowly varying solitary wave asymptotic solution of the perturbed Korteweg-de Vries equation. The well-known results for the variation of the solitary-wave amplitude and the accompanying trailing tail are recovered. Here the analysis is carried through to second order so as to determine a general expression for the first-order speed correction. The result obtained here generalizes and improves previous results.     

The long‐wave limit for the water wave problem I. The case of zero surface tension

The Korteweg‐de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two‐dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long‐wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg‐de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.     

Asymptotic multiphase Σ-solutions to the singularly perturbed Korteweg–de Vries equation with variable coefficients

The paper deals with constructing the asymptotic solution to the singularly perturbed Korteweg–de Vries equation with variable coefficients. The notion of an asymptotic multiphase Σ-solution is proposed, and an algorithm of its construction in a neighborhood of the point t = 0 is given. Some theorems concerning the exactness, with which such local asymptotic solution satisfies the equation under study are proved.     

Large‐time behaviour of solutions to non‐linear wave equations: higher‐order asymptotics

The Cauchy problems for the Korteweg–de Vries–Burgers equation and the Benjamin–Bona– Mahony–Burgers equation are studied. Using subtle estimates of solutions to the linearized equations, the higher‐order terms of the asymptotic expansion as of solutions are derived. Copyright © 1999 John Wiley & Sons, Ltd.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

A note on solutions of the Korteweg–de Vries hierarchy

olutions of the Korteweg–de Vries hierarchy are discussed. It is shown that results by Wazwaz [Wazwaz AM. Multiple-soliton solutions of the perturbed KdV equation. Commun Nonlinear Sci Simul 2010;15911:3270–73] are the well-known consequences of the full integrability for the Korteweg–de Vries hierarchy.     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.