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In this paper, we prove that the modified Camassa–Holm equation is Painlevé integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of weak solutions of the coupled time-dependent Ginzburg-Landau equations and establish the global existence of weak solutions to the equations by Galerkin method and compactness theory. [ABSTRACT FROM AUTHOR]

This paper studies low-regularity solutions of the periodic Fornberg–Whitham equation with initial value. The existence and the uniqueness of solutions are proved. The results are illustrated by considering the periodic peakons of the periodic Fornberg–Whitham equation. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new ten-point multisymplectic scheme for the regularized long-wave equation from its Bridges’ multisymplectic form. The new scheme is an explicit scheme in the sense that it does not need iteration. We discuss some properties of the new scheme. The performance and the efficiency of the new scheme are illustrated by solving several test examples. The obtained results are presented and compared with previous methods. Numerical results indicate that the multisymplectic scheme cannot only obtain satisfied solutions for the regularized long-wave equation but also keep three invariants of motion which are evaluated to determine the conservation properties of the algorithm very well. [ABSTRACT FROM AUTHOR]

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations D0+α(x-x0)=f(t,x) which includes D0+α(x-x0)=H(t)xλ with λ∈(0,1) for the case of slowly decaying coefficients H. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x(t)=o(taα) when t→+∞ for 1>α>1-a>λ>0. Our result can be thought of as a noninteger counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal. [ABSTRACT FROM AUTHOR]

An important subset of distributions encountered in physics is the set of multiplet delta distributions δcy(k), with support a quadratic O(p,q)-invariant surface cy≜{x∈Rn:P(x)=y}, n=p+q. The evaluation of these distributions for a general test function is not always sufficiently detailed in the classical literature, especially for those distributions that are defined as a regularization and/or when one needs their causal or anticausal version (for p=1 or q=1). This work intends to improve this situation by deriving explicit expressions for 〈δcy(k),φ〉, ∀p,q∈Z+, ∀k∈N, ∀y∈R, and ∀φ∈D(Rn), in a form suitable for practical applications. In addition, we also apply to these distributions a new approach to regularization. Distributions that need to be regularized are (equivalently) defined as extensions of a partial distribution. This extension process reveals in a natural way that regularized multiplet delta distributions are in general, uncountably multivalued. In the work of Gel’fand and Shilov [Generalized Functions (Academic, New York, 1964), Vol. 1] four types of multiplet delta distributions with support the null space c0 were introduced: δ1(k)(P), δ2(k)(P) and δ(k)(P+), δ(k)(P-). Our regularization method explains why this particular nonuniqueness was observed and further discloses the full extent of this nonuniqueness. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of Zakharov equations driven by random force is studied. The force, which is smooth enough and homogeneous in space and white noise in time, acts on both equations of the Zakharov system independently. By some a priori estimates we prove the existence of a solution in energy spaces E1 and E2 via the Galerkin approximation. This solution is defined on the given probability space rather than a martingale solution. Then a global random attractor is constructed in energy space E2 equipped with weak topology. Further the existence of a stationary measure is proved in energy space E2 with usual topology. [ABSTRACT FROM AUTHOR]

The properties of strongly gravitating systems suggest that field theory overcounts the states of a system. Reducing the number of degrees of freedom, without abandoning the notion of effective field theory, may be achieved through a connection between the ultraviolet and infrared cutoffs. We provide an implementation of this idea within the Wilsonian approach to the renormalization group. We derive an exact flow equation that describes the evolution of the effective action. We discuss the implications for the existence of infrared fixed points and the running of couplings. We also give an alternative derivation in the context of the perturbative renormalization group. [ABSTRACT FROM AUTHOR]

A fourth-order finite difference solver for the nerve conduction equation is presented that has mesh-independent convergence rate and unconditional convergence for any time-step size. Numerics are performed for a number of test problems which shows that the features of the nerve conduction equation are preserved. The computational loads are roughly same as ones of normal second-order scheme, such as the Crank–Nicolson scheme. [ABSTRACT FROM AUTHOR]

We consider a number of mathematical properties of Maxwell’s equations for linear dispersive and absorptive dielectric media using the auxiliary field method developed earlier by the author [A. Tip, Phys. Rev. A 57, 4818 (1998)]. Here the fields are interpreted as square integrable functions of x∈R3. In case the susceptibility χ(x,t) is piecewise constant in x, we show rigorously that a decomposition into independent equations for longitudinal and transverse fields can be made. We point out its relevance for the study of spectral properties of photonic crystals. Again, for the piecewise constant case we discuss the usual boundary conditions at interfaces and discuss the different nature of those for the longitudinal and transverse fields. Then we consider energy conservation for dispersive, nonabsorptive, media. We show that additional contributions to the free field energy density, as given in the literature, are associated with the energy stored in the auxiliary field modes. Finally, we show that also for nonlinear dielectrics it is possible to obtain a conserved energy by introducing auxiliary fields. [ABSTRACT FROM AUTHOR]