Dynamics of Embedded Solitons in the Extended Korteweg–de Vries Equations

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is based on the development of a soliton perturbation theory. By obtaining the analytical formula for the tail amplitudes of nonlocal solitary waves, we demonstrate the existence of single-hump embedded solitons for both Hamiltonian and non-Hamiltonian perturbations. These embedded solitons can be isolated (existing at a unique wave speed) or continuous (existing at all wave speeds). Under small wave speed limit, our results show that the tail amplitudes of nonlocal waves are exponentially small, and the product of the amplitude and cosine of the phase is a constant to leading order. This qualitatively reproduces the previous results on the fifth-order KdV equation obtained by exponential asymptotics techniques. We further study the dynamics of embedded solitons and prove that, under Hamiltonian perturbations, a localized wave initially moving faster than the embedded soliton will asymptotically approach this embedded soliton, whereas a localized wave moving slower than the embedded soliton will decay into radiation. Thus, the embedded soliton is semistable. Under non-Hamiltonian perturbations, stable embedded solitons are found for the first time.     

Acoustic Scattering and the Extended Korteweg– de Vries Hierarchy

The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa–Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.     

Three-Phase Solutions of the Kadomtsev–Petviashvili Equation

The Kadomtsev–Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N , based on a Riemann theta-function of N variables. For N =1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta-functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real-valued, smooth KP solution associated with a Riemann theta-function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three-phase solutions.     

On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceH^N(S¹; ), (N?1), Hamiltonian (q):=∫_S1((∂_xq(x))²+q(x)³) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN?1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH¹(S¹; ), the KdV-Hamiltonian (and similarly any Hamiltonian in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vectorfield of KdV.     

Generation of Secondary Solitary Waves in the Variable-Coefficient Korteweg–de Vries Equation

We consider the solitary wave solutions of a Korteweg–de Vries equation, where the coefficients in the equation vary with time over a certain region. When these coefficients vary rapidly compared with the solitary wave, then it is well known that the solitary wave may fission into two or more solitary waves. On the other hand, when these coefficients vary slowly, the solitary wave deforms adiabatically with the production of a trailing shelf. In this paper we re-examine this latter case, and show that the trailing shelf, on a very long time-scale, can lead to the generation of small secondary solitary waves. This result thus provides a connection between the adiabatic deformation regime and the fission regime.     

Soliton Interactions of the Kadomtsev–Petviashvili Equation and Generation of Large-Amplitude Water Waves

We study the maximum wave amplitude produced by line-soliton interactions of the Kadomtsev–Petviashvili II (KPII) equation, and we discuss a mechanism of generation of large amplitude shallow water waves by multi-soliton interactions of KPII. We also describe a method to predict the possible maximum wave amplitude from asymptotic data. Finally, we report on numerical simulations of multi-soliton complexes of the KPII equation which verify the robustness of all types of soliton interactions and web-like structure.     

Une redécouverte de l'équation de Korteweg–de Vries et de Kadomtsev–Petviashvili

We show that the flux of long waves of water surface, propagating in each characteristic direction of the equations for a vibrating string, to a first approximation, are close to the solutions of the Korteweg–de Vries equation. In a three dimensional flow, the phenomenon is of the same order as the Kadomtsev–Petviashvili equation. To cite this article: T. Kano, T. Nishida, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Initial Time Layers and Kadomtsev–Petviashvili-Type Equations

The Kadomtsev–Petviashvili (KP) equation and generalizations (GKP) have temporal discontinuities at the initial instant of time. Motivated by the study of water waves, a generalized Boussinesq equation that contains the GKP equations as an "outer" limit is introduced. Within the context of matched asymptotic expansions the discontinuities are resolved. The linear system is analyzed in more detail and the limit process is rigorously established.     

Kadomtsev–Petviashvili Equation Revisited and Integrability in 4 + 2 and 3 + 1

The Cauchy problem of Kadomtsev–Petviashvili I (KPI) was reduced to a nonlocal Riemann–Hilbert (RH) problem by the author and Ablowitz in 1983. This formulation was based on the introduction of two spectral functions (nonlinear Fourier transforms, FTs). This formalism was improved by Boiti et al. [ 1 ], where it was shown that the earlier nonlocal RH problem can be formulated in terms of a single spectral function (nonlinear FT). A different formalism was presented by Zhou [ 2 ], where the Cauchy problem was rigorously solved in terms of a linear integral equation involving a nonanalytic eigenfunction. Here, we first revisit the above results and then review some recent results about the derivation of integrable generalizations of KP in 4 + 2 (i.e., in four spatial and two temporal dimensions), as well as in 3 + 1 (i.e., in three spatial and one temporal dimensions).     

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well‐posedness of the Whitham system.     

Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg–De Vries Equation

The appearance and disappearance of short-lived large-amplitude pulses in a nonlinear long wave model is studied in the framework of the modified Korteweg–de Vries equation. The major mechanism of such wave generation is modulational instability leading to the generation and interaction of the breathers. The properties of breathers are studied both within the modified Korteweg–de Vries equation, and also within the nonlinear Schrödinger equations derived by an asymptotic reduction from the modified Korteweg–de Vries for weakly nonlinear wave packets. The associated spectral problems (AKNS or Zakharov-Shabat) of the inverse-scattering transform technique also are utilized. Wave formation due to this modulational instability is investigated for localized and for periodic disturbances. Nonlinear-dispersive focusing is identified as a possible mechanism for the formation of anomalously large pulses.     

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.     

On the regularization mechanism for the periodic Korteweg–de Vries equation

In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣ^s for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L₂ we also show the Lipschitz‐continuous dependence of these solutions with respect to the initial data as maps from Ḣ^s to Ḣ^s for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.     

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     

Discrete artificial boundary conditions for the linearized Korteweg–de Vries equation

We consider the derivation of continuous and fully discrete artificial boundary conditions for the linearized Korteweg–de Vries equation. We show that we can obtain them for any constant velocities and any dispersion. The discrete artificial boundary conditions are provided for two different numerical schemes. In both continuous and discrete case, the boundary conditions are nonlocal with respect to time variable. We propose fast evaluations of discrete convolutions. We present various numerical tests which show the effectiveness of the artificial boundary conditions.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1455–1484, 2016