Novel rogue waves in an inhomogenous nonlinear medium with external potentials

Employing the similarity transformation connected with the standard constant coefficient nonlinear Schrödinger equation, we obtain the analytical rogue wave solutions to a generalized variable coefficient nonlinear Schrödinger equation with external potentials describing the pulse propagation in nonlinear media with transverse and longitudinal directions nonuniformly distributed. Based on the obtained solutions, abundant structures of rogue waves are constructed by selecting some special parameters. The main properties as well as the dynamic behaviors of these rogue waves are discussed by direct computer simulations.     

Standing waves in an open pipe: A nonlinear initial-boundary value problem

Summary Transient disturbances in a column of gas in an open tube are described using a nonlinear theory which includes both amplitude dispersion and wave interactions. For reflection from an open end, the theory must include the second nonlinear correction to the characteristic to distinguish the linear and nonlinear travel times. The initial value problem is reduced to solving a functional difference equation which determines the signal on a boundary. When the signal is damped by allowing radiation of energy at the boundaries, it may be possible to prevent shock formation.

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Zusammenfassung Zeitabhängige Störungen in einem mit Gas gefüllten offenen Rohr werden beschrieben mit Hilfe einer nicht-linearen Theorie, welche sowohl Amplituden-Dispersion wie auch gegenseitige Beeinflussung von Wellenzügen berücksichtigt. Für Reflektionen am offenen Ende muss die Theorie die nicht-linearen Korrekturen zweiter Ordnung zu den Charakteristiken berücksichtigen, um lineare und nicht-lineare Laufzeiten unterscheiden zu können. Die Anfangswertaufgabe wird auf die Lösung einer Funktional-Differenzen-Gleichung reduziert, welche das Signal am Rohrende bestimmt. Wenn das Signal gedämpft wird durch Zulassung von Abstrahlung am Rohrende, kann die Stossbildung vermieden werden.

     

The direct and inverse problem for an inclusion within a heat-conducting layered medium

This paper is concerned with the problem of heat conduction from an inclusion in a heat transfer layered medium. Making use of the boundary integral equation method, the well-posedness of the forward problem is established by the Fredholm theory. Then an inverse boundary value problem, i.e. identifying the inclusion from the measurements of the temperature and heat flux on the accessible exterior boundary of the medium is considered in the framework of the linear sampling method. Based on a careful analysis of the Dirichlet-to-Neumann map, the mathematical fundamentals of the linear sampling method for reconstructing the inclusion are proved rigorously.     

Propagation of nonlinear magnetoacoustic waves in a compressible medium of finite electrical conductivity

We study the one-dimensional propagation of weakly nonlinear waves in a compressible medium of finite electrical conductivity subjected to the action of a magnetic field. We obtain evolution equations that describe the wave processes under small and finite magnetic Reynolds numbers. It is shown that in a medium of finite conductivity the evolution of perturbations in a fluid is described by the modified Bürger's equation. We find the stationary and automodel solution of this equation and use them as the basis for analyzing the influence of effects of electrical conductivity on the structure of weak shock waves.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 73–76.     

Propagation of weakly guided waves in a Kerr nonlinear medium using a perturbation approach

The equations are represented in a simplified format with only a few leading terms needed in the expansion. The set of equations are then solved numerically using vector finite element method. To validate our algorithm, we analyzed a two-dimensional rectangular waveguide consisting of a linear core and nonlinear identical cladding. The exact nonlinear solutions for three different modes of propagations, TE0, TE1, and TE2 modes are generated and compared with the computed solutions. Next, we investigate the effect of a more intense monochromatic field on the propagation of a “weak” optical field in a fully three-dimensional cylindrical waveguide.     

Space-time ray method for waves of small deformation in a nonlinear elastic medium

Space-time ray solutions for longitudinal waves of small deformation in a non-linear elastic medium are constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 61–72, 1984.     

Shock wave structure in an inviscid heat-conducting locally equilibrium medium of a perfect gas with radiation

The structure of a shock wave (SW) in an inviscid heat-conducting locally equilibrium medium of a perfect gas with radiation at temperatures up to hundreds of thousands and millions of degrees is analysed in the approximation of non-linear heat conduction. At such temperatures, heat conduction is far more important than viscosity, the medium consists of ions, electrons and radiation, and the electrons and radiation make a decisive contribution to the heat transfer and an appreciable contribution to the thermodynamic functions of the medium. Combinations with the dimensions of all the flow parameters are constructed from the gas density ahead of the SW and the constants appearing in the equations of state of such a medium. If they are taken as the corresponding scales, the equations of state for different media in dimensionless form only differ in the ratio of the specific heat capacities (the adiabatic exponent) of the gas. The SW structure is studied for adiabatic exponents from 1 to 3, dimensionless temperatures ahead of the SW from zero to infinity and any SW velocities exceeding the speed of sound in the unperturbed medium. The following cases are successively considered: 1) when the contribution due to radiation is neglected in the equations of state, 2) when radiation is taken into account and the SW moves through a gas at a zero temperature (through a cold background), and 3) the radiation contribution is allowed for in the case of a warm background. It is ascertained when the SW structure is continuous and when it contains a finite or infinite forerunner and an isothermal shock. The transition from a continuous structure to a structure with an isothermal shock and the intensity of this shock are independent of the form of the formulae for the thermal conductivity in Fourier's law. In the case of adiabatic exponents greater than unity, the structure is continuous for all SW velocities starting from a certain dimensionless background temperature.     

On a new class of nonlinear wave equations

In this paper we prove the existence and uniqueness of regular solutions for the Cauchy problem for the evolution equation $\text{u″ + A}{}^2\text{u + (α + M(¦A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u¦}{}^2\text{) Au = 0}$, suggested by the study of beams and plates. We represent by A a linear operator of a Hilbert space H with norm ∥, α is a real number, and M(λ) > 0 a real function, for λ ? 0.     

Numerical solution of nonlinear equations by an optimal eighth-order class of iterative methods

Solving nonlinear equations by using iterative methods is discussed in this paper. An optimally convergent class of efficient three-point three-step methods without memory is suggested. Analytical proof for the class of methods is given to show the eighth-order convergence and also reveal its consistency with the conjecture of Kung and Traub. The beauty in the proposed methods from the class can be seen because of the optimization in important effecting factors, i.e. optimality order, lesser number of functional evaluations; as well as in viewpoint of efficiency index. The accuracy of some iterative methods from the proposed derivative-involved scheme is illustrated by solving numerical test problems and comparing with the available methods in the literatures.     

Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations

This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p ＋ q 〈 4/N. By using the variational calculus and scaling argument, we establish the existence of standing waves with ground state, discuss the behavior of standing waves as a function of the frequency ω and give the sufficient conditions of the stability of the standing waves with the least energy for the equations under study.     

Instability of standing waves for a class of nonlinear Schrödinger equations

This paper discusses a class of nonlinear Schrödinger equations with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blowup and global existence to the solutions of the Cauchy problem and answer such a problem: how small are the initial data, the global solutions exist? At last we prove the instability of standing wave by combing those results.     

Hyperbolic equations of Maxwell's nonlinear model of elastoplastic heat-conducting media

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 135–159, July–August, 1989.     

Diffraction in a nonlinear defocusing medium

Diffraction by a semitransparent screen is considered in the framework of the nonlinear Schrödinger equation. The case of a defocusing medium is investigated. The method used represents a synthesis of the method of Riemann's matrix problem and the technique of Whitham's deformations of spectral curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 23–31, 1989.     

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.     

A new approach for solving a class of nonlinear integro-differential equations

In this paper, a new technique for solving a class of quadratic integral and integro-differential equations is introduced. The main advantage of this technique is that it can replace the nonlinear problem by an equivalent linear one or by another simpler nonlinear one. The convergence of the series solution is proved. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. Some numerical examples are introduced to verify the efficiency of the new technique.