Exact explicit traveling wave solutions for two nonlinear Schrödinger type equations

The traveling wave solutions of the generalized nonlinear derivative Schrödinger equation and the high-order dispersive nonlinear Schrödinger equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions) are given.     

Chern–Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field

We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern–Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)–(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant q goes to infinity.     

Long Range Scattering and Modified Wave Operators for the Wave-Schrödinger System II

We continue the study of scattering theory for the system consisting of a Schrödinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper, we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the different propagation properties of the wave and Schrödinger equations. Here we eliminate that condition by using an improved asymptotic form for the solutions. Communicated by Bernard Helffer submitted 20/02/03, accepted: 24/06/03     

Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrödinger equation with time-dependent nonlinearity

In this paper, we consider a general form of nonlinear Schrödinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schrödinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.     

Matrix technique for nonperturbative Green function of time-independent Schrödinger equation

A Green function of time-independent multi-channel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multi-channel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multi-channel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multi-potential system within quasi-classical approximation. The limits of strong and weak inter-channel interactions are studied.     

Weighted energy decay for 1D wave equation

We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation.     

Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

Butterfly-like waves in the nonlinear Schrödinger equation with a combined dispersion term

All possible optical waves in the nonlinear Schrödinger equation with a combined dispersion term are determined according to different parameters’ regions. First, we find this equation has some popular exotic solutions, such as peaked waves, looped and cusped waves. What is more interesting, this equation admits some very particular waves, such as double kinked waves and butterfly-like waves. These new two types of solutions have not been reported in the literature regarding the study of other nonlinear equations.     

Exact solutions to nonlinear Schrödinger equation with variable coefficients

According to Ma-Fuchsseiter’s idea, a trial equation method was proposed to find the exact envelop traveling wave solutions to some nonlinear differential equations with variable coefficients. As an application, combining with the complete discrimination system for polynomial, some exact envelop traveling wave solutions to Schrödinger equation with variable coefficients were obtained. At the same time, the physical meanings of the obtained solutions are discussed, and the problem needed to further study is pointed out.     

Discrete exact solutions to some nonlinear differential-difference equations via the (G′/G)-expansion method

We extended the (G′/G)-expansion method to two well-known nonlinear differential-difference equations, the discrete nonlinear Schrödinger equation and the Toda lattice equation, for constructing traveling wave solutions. Discrete soliton and periodic wave solutions with more arbitrary parameters, as well as discrete rational wave solutions, are revealed. It seems that the utilized method can provide highly accurate discrete exact solutions to NDDEs arising in applied mathematical and physical sciences.     

Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine-cosine and the exp-function methods

In this article we find the exact traveling wave solutions of the generalized nonlinear Schrödinger (GNLS) equation with variable coefficients using three methods via the generalized extended tanh-function method, the sine-cosine method and the exp-function method. The main objective of this article is to compare the efficiency of these methods by delivering the exact traveling wave solutions of the proposed nonlinear equation.     

Generalized Eigenfunctions for Waves in Inhomogeneous Media

Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrödinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrödinger form, leading to the study of the spectral theory of its classical wave operator, a self-adjoint, partial differential operator on a Hilbert space of vector-valued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure.     

Kato-type estimates for NLS equation in a scalar field and unique solvability of NKGS system in energy space

We consider Schrödinger equation in R²⁺¹${\mathbb{R}}^{2+1}$ with nonlinear scalar potential. The potentials are time-independent or determined as solutions to inhomogeneous wave equations. By constructing a modified propagator, we derive Kato-type smoothing estimates for the nonlinear Schrödinger (NLS) equation. With the help of these results, we prove the unique solvability of the nonlinear Klein–Gordon–Schrödinger (NKGS) system for all time in the energy space.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

Exact analytic solutions of Schrödinger linear and nonlinear equations

Exact analytic solutions of Schrödinger linear partial differential equations are obtained. Moreover, the cubic nonlinear Schrödinger equation is treated with the use of a well-known functional analytic method and the existence of convergent power series solutions is proved. From these solutions, under certain initial conditions, similar results as those presented in the literature are obtained.     

Discrete Spectrum of a Three-Particle Schrodinger Operator with a Homogeneous Magnetic Field

The discrete spectrum of the Schrödinger operator is studiedfor a system of three identical particles with short-range interactionsin a homogeneous magnetic field. All the two-particle subsystemsare supposed to be unstable. Finiteness of the discrete spectrumis established under some assumptions about the solutions ofthe corresponding two-particle Schrödinger equation.     

Non-Lie ansatzen and conditional symmetry of the nonlinear Schrödinger equation

A new approach to the construction of non-Lie ansatzen that reduce the multidimensional nonlinear Schrödinger equation to ordinary differential equations is proposed. Besides the well-known solutions that may be obtained by means of Lie symmetry, other solutions are found that may be generated by means of conditional invariance operators of the Schrödinger equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1620–1628, December, 1991.     

Boundary-value problem for the nonlinear Schrödinger equation

A mixed boundary-value problem for the nonlinear Schrödinger equation and its generalization is studied by the method used for the inverse scattering problem. A connection is established between conservation laws and boundary conditions in integrable boundary-value problems for higher nonlinear Schrödinger equations. It is shown that the generalized boundary-value problem requires a joint consideration of regular and singular solutions for the nonlinear Schrödinger equation with repulsion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 151–165, 1988.     

On plane wave solutions of non-symmetric field equations of unified theories of Einstein,Bonnor and Schrödinger

Summary Various attempts have been made to obtain wave solutions of nonsymmetric unified field equations of Einstein, Bonnor and Schrödinver in different space-times. Lal and Ali, Lal and Khare, Lal and Singh have obtained wave solutions of these field equations in different space-times under certain restrictions. In this paper plane wave solutions of both the weak and strong, nonsymmetric unified field equations of Einstein, Bonnor and Schrödinger have been investigated in a space which is more general than Pandey and Rao plane wave metric. It has been shown that the plane wave solutions of Einstein and Bonnor's equations exist but Schrödinger's field equations have no solution in this space-time.     

On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.