Local Well-Posedness for a Fourth-Order Nonlinear Dispersive System on the 1D Torus

The stability of various equilibrium solutions of a strongly dispersive nonlinear system with instability and dissipation is investigated both numerically and analytically. Periodic trains of soliton-like pulses are found to be stable when the distance between adjacent pulses becomes smaller than a critical value. This critical value is determined by linear stability analysis. A modulational type instability is also observed for a very long string of soliton-like pulses even when the fundamental distance is within the stable regime.

Calculations of a discrete nonlinear dispersive wave system show that as the degree of nonlinearity increases, the system experiences in turn, periodic, recurring, chaotic, transitional, and periodic motions. A relationship between the instability of the initial configuration and the long-time behavior is identified. The calculations further suggest that the corresponding continuous system will exhibit chaotic motions and energy-sharing among a narrow band of unstable modes, a phenomenon which we call ‘‘confined chaos.’’

This paper proposes a numerical scheme for the (2 + 1)‐dimensional nonlinear Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation (ZK‐BBME). The ZK‐BBME represents a long‐wave model with large wavelength that explains the water wave phenomena in nonlinear dispersive system. The solution of the ZK‐BBME is discretized by using hybridization of the finite difference and localized radial basis function based on partition of unity method. The association of this hybridization leads to a meshless method and it does not requires any linearization. In the initial step, the PDE is converted into a nonlinear ODE system by making use of radial kernels. In the next step, the obtained nonlinear ODE system is solved based on an ODE solver of high order. The global collocation technique imposes a large computational cost because a dense algebraic system must be calculated. This proposed strategy is based on decomposing the initial domain into a number of sub‐domains using kernel approximation on each sub‐domain. Three numerical examples are illustrated to clarify the efficiency and accuracy of the proposed strategy.

A. We establish local well-posedness and smoothing results for the Cauchy problem of a degenerate dispersive Navier-Stokes system that arises from kinetic theory. Under assumptions that the initial data satisfy asymptotic flatness and nontrapping conditions, we show there exists a unique classical solution for a finite time. Due to degeneracies in both dissipation and dispersion for the system, different components of the solution gain different regularity. Couplings of these components are analyzed using pseudodifferential operators.

We consider the Serre system of equations, a nonlinear dispersive system that models two-way propagation of long waves of not necessarily small amplitude on the surface of an ideal fluid in a channel. We discretize in space the periodic initial-value problem for the system using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove an optimal-order $L^{2}$-error estimate for the resulting semidiscrete approximation. Using the fourth-order accurate, explicit, "classical" Runge--Kutta scheme for time stepping, we construct a highly accurate fully discrete scheme in order to approximate solutions of the system, in particular, solitary-wave solutions, and study numerically phenomena such as the resolution of general initial profiles into sequences of solitary waves, and overtaking collisions of pairs of solitary waves propagating in the same direction with different speeds.

Determination of boundaries of unsteady oscillatory zone in asymptotic solutions of non-integrable dispersive wave equations

The initial-boundary value problem for a nonlinear dispersive system with time-dependent boundary condition is discussed in the Sobolev space ${H^1}$ from the point of view of the theory of nonlinear evolution operators. A notion of weak solution to the problem is introduced and the associated family of solution operators is constructed in such a way that it gives rise to a nonlinear evolution operator with time-dependent domain. Various qualitative properties as well as regularity of the weak solutions are investigated through those of the constructed evolution operator.

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter \alpha > 1 , subject to a Gaussian random initial data of negative Sobolev regularity \sigma<s-{1}/{2} , for s \le 1/2 . We show that for all s_{\ast}(\alpha) <s\leq{1}/{2} , the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on H^{s}(\mathbb{T}) is quasi-invariant under the flow of the equation. For \alpha < \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537 , the regularity of the initial data is lower than the one provided by the deterministic well-posedness theory. This is the first probabilistic globalization argument that is not in the setting of an invariant measure and not based on a known deterministic method to construct global-in-time solutions. We obtain this result by following the approach of DiPerna–Lions (1989), first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain’s invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from the probabilistic local well-posedness theory.

Historically, the study of nonlinear dispersive waves started with the pioneering work of Stokes in 1847 on water waves. Stokes first proved the existence of periodic wavetrains which are possible in nonlinear dispersive wave systems. He also determined that the dispersion relation on the amplitude produces significant qualitative changes in the behavior of nonlinear waves. It also introduces many new phenomena in the theory of dispersive waves, not merely the correction of linear results. These fundamental ideas and the results of Stokes have provided a tremendous impact on the subject of nonlinear water waves, in particular, and on nonlinear dispersive wave phenomena, in general. Stokes’ profound investigations on water waves can be considered as the starting point for the modern theory of nonlinear dispersive waves. In fact, most of the fundamental concepts and results on nonlinear dispersive waves originated in the investigation of water waves. The study of nonlinear dispersive waves has proceeded at a very rapid pace with remarkable developments over the past three decades.

An analytical solution is obtained for solitary pulse propagation in an amplified nonlinear dispersive system. For a homogeneously broadened gain medium, this solitary pulse has a hyperbolic secant amplitude and a hyperbolic tangent instantaneous frequency variation. The pulse is a gain-guided pulse in either the positive or the negative dispersion regime as well as in the self-focusing or self-defocusing regime. A dark solitary pulse that has a hyperbolic tangent amplitude and a similar instantaneous frequency variation is also obtained.

The primary recurrent theme in ‘soliton physics’ as it is now evolving [1] in statistical physics, critical phenomena and condensed-matter physics generally (as well as quantum-field theory and gravity, etc.), is the simple recognition that linearization schemes are frequently misleading and (as witnessed by the contents of this Symposium) sometimes unnecessary. It is increasingly appreciated that we need to go beyond (linear) normal modes and finite-order perturbation theories in a spectrum of intrinsically nonlinear problems for which distinctive nonlinear, e.g., soliton, sectors of solution space are possible. Since the physical signatures of nonlinear modes, particularly spatially-limited ones, are often so crucial, it is essential that they are fully represented. In this situation, there is a growing tendency to develop theories which ab initio include all fundamental modes (including nonlinear ones) even if this is only possible semi-quantitatively. Such an approach has well-established precedents, e.g., phonon-roton gas models in liquid-helium-4 or vortices in superconductors.