Generalized KdV Equation Research Articles

Analyzing coupled-wave dynamics: lump, breather, two-wave and three-wave interactions in a (3+1)-dimensional generalized KdV equation

Analyzing coupled-wave dynamics: lump, breather, two-wave and three-wave interactions in a (3+1)-dimensional generalized KdV equation

Bilinear Bäcklund transformation, Lax pair, Painlevé integrability, and different wave structures of a 3D generalized KdV equation

Bilinear Bäcklund transformation, Lax pair, Painlevé integrability, and different wave structures of a 3D generalized KdV equation

Local existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p-generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions:c×⋯×c︸p(total distance):=∑♣1,⋯,♣p∈Zν♣1+⋯+♣p=total distance∏j=1pc(♣j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p=2[11], and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p=3. Next, we prove existence and uniqueness in the general case p≥2, which then covers the remaining cases p≥4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p.

Spatial analyticity of solutions for a coupled system of generalized KdV equations

We consider the well‐posedness of a coupled system for generalized Korteweg–de Vries (S‐gKdV) where and the initial data are analytic in a strip and then the solutions of S‐gKdV continue to be analytic in a strip the width of which will decrease as time goes to , by using the standard contraction method in analytic Bourgain space. Besides, we obtain algebraic lower bounds on the decreasing rate of the uniform radius of analyticity of the solutions.

Small Data Scattering for the Global Solutions of the Supercritical Generalized KdV Equation

We consider the scattering problem for the global solutions of the supercritical generalized KdV equation ∂tu+∂xxxu+μ∂x(uk+1) = 0, where k > 4 is an integer, initial data u0 belongs to H1(ℝ), and μ = ±1. To solve the scattering problem, a scattering criteria is established firstly, and then a new inequality is introduced to obtain uniformly bounded solutions in H1(ℝ). Finally, we further clarify the conditions for the equation to have a global solution scattering in H1(ℝ). Our method is mainly inspired by the works of Farah, Linares, Pastor, and Visciglia.

Space-curved resonant solitons and inelastic interaction solutions of a (2+1)-dimensional generalized KdV equation

Space-curved resonant solitons and inelastic interaction solutions of a (2+1)-dimensional generalized KdV equation

Improved algebraic lower bound for the radius of spatial analyticity for the generalized KdV equation

We consider the initial value problem (IVP) for the generalized Korteweg–de Vries (gKdV) equation ∂tu+∂x3u+μuk∂xu=0,x∈R,t∈R,u(x,0)=u0(x),where u(x,t) is a real valued function, u0(x) is a real analytic function, μ=±1 and k≥4. We prove that if the initial data u0 has radius of analyticity σ0, then there exists T0>0 such that the radius of spatial analyticity of the solution remains the same in the time interval [−T0,T0]. In the defocusing case, for k∈2N, k≥4, we prove that when the local solution extends globally in time, then for any T≥T0, the radius of analyticity cannot decay faster than cT−2kk+4+ϵ, ϵ>0 arbitrarily small and c>0 a constant. The result of this work improves the one by Bona et al. (2005) where the authors proved the decay rate is no faster than cT−(k2+3k+2).

Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics

Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics

Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power

Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power

Generalized KdV and Burger Equations and Various Vanishing Limits

In this analytical and numerical study, we look at the generalized KdV equation by letting different coefficients go to zero. Numerical study is carried out by the pseudospectral method. Our study shows that there is no difference between the behavior of the solutions in the limiting case and the solutions in the case when the corresponding coefficients are exactly zero.

Using Vector-Product Loop Algebra to Generate Integrable Systems

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations.

Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data

Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data

Formula omitted]-soliton solutions and associated integrability for a novel (2+1)-dimensional generalized KdV equation

In this paper, we investigate the integrability of a (2+1)-dimensional generalized KdV equation. In virtue of the Weiss–Tabor–Carnevale method and Kruskal ansatz, this equation can pass the Painlevé test. The truncated Painlevé expansion leads to the Bäcklund transformation and rational solutions. The bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are constructed with the Hirota bilinear method and Bell polynomials. It is proved that the (2+1)-dimensional generalized KdV equation can be regarded as an integrable model in sense of infinite conservation laws. The formula of N-soliton solutions is given and verified with the Hirota condition. The study of integrability provides theoretical guidance for solving equations and gives the possibility of the existence of exact solutions.

Nonlocal symmetries and interaction solutions for the (n + 1)-dimensional generalized Korteweg–de Vries equation

The (n + 1)-dimensional generalized KdV equation is presented in this paper, and we further investigate its nonlocal symmetries by different methods. It can be seen that the symmetrical transformations obtained by different nonlocal symmetries are usually equivalent. Based on the obtained Lie point symmetry as well as the mth finite symmetrical transformations, we obtain its soliton molecules and multiple soliton solutions. Additionally, for the case of n = 4 various types of interaction solutions among solitons and periodic waves are obtained by the similarity reduction method.

Non-singular multi-complexiton wave to a generalized KdV equation

Non-singular multi-complexiton wave to a generalized KdV equation

Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity

<p style='text-indent:20px;'>The paper is concerned on solitary waves for singularly perturbed generalized KdV equation with high order nonlinear terms. We firstly give the phase portraits of system related to the unperturbed equation under various cases by theory of planar dynamical system. Then by using geometric singular perturbation theory and Melnikov's method, the existence of solitary wave solutions of generalized KdV equations with high order nonlinear terms is established. It is proven that some solitary wave solutions with particular wave speeds will persist under small perturbations.</p>

Limit speed of traveling wave solutions for the perturbed generalized KdV equation

<p style='text-indent:20px;'>The existence of solitary waves and periodic waves for a perturbed generalized KdV equation is established by using geometric singular perturbation theory. It is proven that the limit wave speed <inline-formula><tex-math id="M1">\begin{document}$ c_{0}(h) $\end{document}</tex-math></inline-formula> is decreasing by analyzing the ratio of Abelian integrals for <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula>. The upper and lower bounds of the limit wave speed are given. Moreover, the relation between the wave speed and the wavelength of traveling waves is obtained. Our results answer partially an open question proposed by Yan, Liu and Liang [Math. Model. Anal., 19 (2014), pp. 537-555].</p>

The invariant ion-acoustic waves in the plasma

The space plasmas have been found empirically to be separated into those residing far from the classical thermal equilibrium and those residing near equilibrium. The modern formalism of the kappa distributions explains this distinction under the value of the kappa index, the intensive parameter that characterizes thermodynamics together with temperature. Recent studies have suggested that by defining an invariant kappa index as the zero dimensionality spectral index, kappa _{0}, which is independent of the dimensionality, the degrees of freedom, or the numbers of particles, one may separately consider the physical and thermodynamic feature of the kappa index in space plasmas by utilizing kappa _{0}. This study extends the mentioned idea to the ion-acoustic waves (IAWs) in the astrophysical plasmas in order to deriving an invariant formalism for the IAWs including the pure thermodynamic features of the background particles. This paper is based on the kinetic theory formalism and the hydrodynamic fluid description for extracting the characteristics of the invariant IAWs. Relying on the Vlasov–Poisson equations, considering a low-frequency band for the weakly damped ion oscillations, we have derived the most generalized formalism of the ion-sound speed in space plasmas in terms of the extended polytropic indices of the plasma species, gamma _{j}, and also the generalized formalism of Landau damping for the invariant IAWs in terms of kappa _{0}, wavelength, and temperatures of the plasma species. In the hydrodynamic description, we have normalized the fluid parameters in terms of the generalized quantities, including the extended formulations of the ion-sound speed and Debye length. Then, by using the perturbation expansion in linear and nonlinear regimes, we may find some other issues in the formalism of the invariant IAWs, such as the effect of the perturbed potential degrees of freedom, d_{Phi }, the isothermal/extended phase speed of the IAWs, and the combined effects of the wave steepening and dispersion of ion waves. We have also derived a generalized KdV equation and its solitary wave solutions in an invariant formalism. Based on the empirical evidences in space plasmas, the far-equilibrium plasmas are characterized by 0<kappa _{0}<1 (0<gamma _{j}<0.5), while the near-equilibrium plasmas are labeled with kappa _{0}>1 (0.5<gamma _{j}<1). We have numerically analyzed our solutions from the anti-equilibrium states at kappa _{0}rightarrow 0 (gamma _{j}rightarrow 0) towards the equilibrium states at kappa _{0}gg 1 (gamma _{j}rightarrow 1). Our theoretical study provides strong evidence, for the first time, about the distinction of plasmas under the value of the kappa index. Our analysis confirms the distinction of the involved IAWs diagrams in the two mentioned regions, where the transition from far-equilibrium states to the near-equilibrium states may occur in the vicinity kappa _{0}sim 1 (gamma _{j}sim 0.5), denoting the escape state of the evolution.

Modified energies for the periodic generalized KdV equation and applications

We construct modified energies for the generalized KdV equation. As a consequence, we obtain quasi-invariance of the high-order Gaussian measures, along with L^q regularity on the corresponding Radon–Nikodym density, as well as new bounds on the growth of the Sobolev norms of the solutions.

Wronskian $$\pmb {N}$$-soliton solutions to a generalized KdV equation in ($$\pmb {2+1}$$)-dimensions

Wronskian $$\pmb {N}$$-soliton solutions to a generalized KdV equation in ($$\pmb {2+1}$$)-dimensions