Correction to: Periodic Waves in the Fractional Modified Korteweg–de Vries Equation

Journal of Dynamics and Differential Equations - A correction to this paper has been published: https://doi.org/10.1007/s10884-021-10016-2     

The Modulational Instability for a Generalized Korteweg–de Vries Equation

We study the spectral stability of a family of periodic standing wave solutions to the generalized Korteweg–de Vries in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evans function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first, an orientation index, counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second, a modulational instability index, provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Hǎrǎguş and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator—in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests a geometric interpretation attached to the vanishing of the orientation index previously mentioned.     

Boundary control of the generalized Korteweg–de Vries–Burgers equation

This paper considers the boundary control problem of the generalized Korteweg–de Vries–Burgers (GKdVB) equation on the interval [0, 1]. We derive a control law of the form and α is a positive integer, and prove that it guarantees L ²-global exponential stability, H ¹-global asymptotic stability, and H ¹-semiglobal exponential stability. Numerical results supporting the analytical ones for both the controlled and uncontrolled equations are presented using a finite element method.     

Robust stabilization the Korteweg–de Vries–Burgers equation by boundary control

The problem of robust global stabilization by nonlinear boundary feedback control for the Korteweg–de Vries–Burgers equation on the domain [0,1] is considered. The main purpose of this paper is to derive nonlinear robust boundary control laws which make the system robustly globally asymptotically stable in spite of uncertainty in the system parameters. Furthermore, we show that the proposed boundary controllers guarantee L ₂-robust exponential stability, L _∞-robust asymptotic stability and boundedness in terms of both L ₂ and L _∞.     

Smooth positon solutions of the focusing modified Korteweg–de Vries equation

The n-fold Darboux transformation \(T_{n}\) of the focusing real modified Korteweg–de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues \(\lambda _{j}\) and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit \(\lambda _{j}\rightarrow \lambda _{1}\), from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding “phase shifts” of the multi-positons are also investigated.     

Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

In this paper, we consider the boundary control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three control laws are derived for this equation and the L ²-global exponential stability of the solution is proved analytically. Numerical results using the finite element method (FEM) are presented to illustrate the developed control schemes.     

Adaptive boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

This paper deals with the adaptive control problem of the unforced generalized Korteweg?Cde Vries?CBurgers (GKdVB) equation when the spatial domain is [0,1]. Three adaptive control laws are designed for the GKdVB equation when either the kinematic viscosity ?? or the dynamic viscosity ?? is unknown, or when both viscosities ?? and ?? are unknowns. Using the Lyapunov theory, the L ²-global exponential stability of the solutions of this equation is shown for each of the proposed control laws. Also, numerical simulations based on the Finite Element method (FEM) are given to illustrate the analytical results.     

A stability analysis of periodic solutions to the steady forced Korteweg–de Vries–Burgers equation

The stability of periodic solutions to the steady forced Korteweg–de Vries–Burgers (fKdVB) equation is investigated here. This family of periodic solutions was identified by Hattam and Clarke (2015) using a multi-scale perturbation technique. Here, Floquet theory is applied to the governing equation. Consequently, two criteria are found that determine when the periodic solutions are stable. This analysis is then confirmed by a numerical study of the steady fKdVB equation.     

Stability of Standing Waves for the Nonlinear Fractional Schrödinger Equation

We study the standing waves of the nonlinear fractional Schrödinger equation. We obtain that when \(0<\gamma <2s\), the standing waves are orbitally stable; when \(\gamma =2s\), the ground state solitary waves are strongly unstable to blow-up.     

Head-on collision of solitary waves in coupled Korteweg–de Vries systems modeling nonlinear transmission lines

The extended Poincaré–Lighthill–Kuo (PLK) method is applied to characterize head-on collisions of solitary waves in a coupled Korteweg–de Vries (KdV) system that has multiple modes supporting solitons. As a simple physically realizable system, we investigate two coupled electrical nonlinear transmission lines (NLTLs), and the proposed method successfully leads to the collision-induced phase shifts and the wave equation that governs the dynamics of the pulses generated by colliding solitary waves.     

An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.     

Wronskian solutions and integrability for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids

Under investigation in this paper is a generalized variable-coefficient forced Korteweg–de Vries equation, which can describe the shallow-water waves, internal gravity waves, and so on. With symbolic computation, the soliton solutions in the Wronskian form are derived based on the given bilinear form. Bäcklund transformation and Lax pair for such equation are also constructed. Variable coefficients and parameters of three solitons are managed to observe the features of the solitonic propagation and interaction, e.g., the solitonic velocity, amplitude and background. Our results could be expected to benefit the relevant problems in fluids.     

Identifying Space-Dependent Coefficients and the Order of Fractionality in Fractional Advection–Diffusion Equation

Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the advection–diffusion equation supplemented by a time derivative of non-integer order. We are describing an inverse method for this equation: It finds the order of the fractional derivative and the coefficients that achieve minimum discrepancy between solution and tracer data. Using an adjoint equation divides the computational effort by an amount proportional to the number of freedom degrees, which becomes large when some coefficients depend on space. Method accuracy is checked on synthetical data, and applicability to actual tracer test is demonstrated.     

Bounded Domain Problem for the Modified Buckley–Leverett Equation

The focus of the present study is the modified Buckley–Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley–Leverett (BL) equation by including a balanced diffusive–dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers a non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that for the MBL equation, the solution of the finite interval \([0,L]\) boundary value problem converges to that of the half line \([0,+\infty )\) boundary value problem exponentially as \(L\rightarrow +\infty \) . This result provides a justification for the use of the finite interval in numerical studies for the half line problem [Y. Wang and C.-Y. Kao, Central schemes for the modified Buckley–Leverett equation, J. Comput. Sci. 4(1–2), 12 – 23, 2013]. Furthermore, we numerically verify that the convergence rate is consistent with the theoretical derivation. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.     

The interpolating element-free Galerkin method for solving Korteweg–de Vries–Rosenau-regularized long-wave equation with error analysis

Nonlinear Dynamics - The main target of this investigation is to develop a new numerical method for solving a class of wave models, i.e., the Korteweg–de Vries–Rosenau-regularized...     

Periodic and Solitary Travelling Waves in Infinite Lattices without Ambrosetti–Rabinowitz Condition

In this paper, we consider FPU lattices with particles of unit mass. The dynamics of the system is described by the infinite system of second order differential equations

$$\begin{aligned} \ddot{q}_n= U^{\prime }(q_{n+1}-q_n)-U^{\prime }(q_n-q_{n-1}),\quad n\in \mathbb {Z}, \end{aligned}$$

where \(q_n\) denotes the displacement of the \(n\)-th lattice site and \(U\) is the potential of interaction between two adjacent particles. We investigate the existence of two kinds travelling wave solutions: periodic and solitary ones under some growth conditions on \(U\) which is different from the widely used Ambrosetti–Rabinowitz condition.

     

Nonlocal symmetry and similarity reductions for a $$\varvec{}$$-dimensional Korteweg–de Vries equation

Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.     

Employing the dynamics of poles in the complex plane to describe properties of rogue waves: case studies using the Boussinesq and complex modified Korteweg–de Vries equations

Nonlinear Dynamics - The dynamics and properties of rogue waves of two classical evolution equations are studied in terms of trajectories of the poles of the exact solutions, by analytically...     

Time Periodic Traveling Curved Fronts in the Periodic Lotka–Volterra Competition–Diffusion System

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$

where \(\Delta \) denotes \(\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }\), \(x,y\in {\mathbb {R}}\) and \(d>0\) is a constant, the functions \(r_i(t),a_i(t)\) and \(b_i(t)\) are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution \(\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) \) connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.

     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.