A discrete negative AKNS equation: generalized Cauchy matrix approach

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit we obtain a semidiscrete negative AKNS equation and after a second continuum limit we derive the nonlinear negative AKNS equation. The reductions to discrete, semi-discrete and continuous sine-Gordon equations are also discussed.     

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.     

Darboux Transformation for a Negative Order AKNS Equation

Using a quasideterminant Darboux matrix, we compute soliton solutions of a negative order AKNS (AKNS($-$1)) equation. Darboux transformation (DT) is defined on the solutions to the Lax pair and the AKNS($-$1) equation. By iterated DT to K-times, we obtain multisoliton solutions. It has been shown that multisoliton solutions can be expressed in terms of quasideterminants and shown to be related with the dressed solutions as obtained by dressing method.     

On a generalized Ablowitz–Kaup–Newell–Segur hierarchy in inhomogeneities of media: soliton solutions and wave propagation influenced from coefficient functions and scattering data

In this paper, a generalized Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy in inhomogeneities of media described by variable coefficients is investigated, which includes some important nonlinear evolution equations as special cases, for example, the celebrated Korteweg–de Vries equation modeling waves on shallow water surfaces. To be specific, the known AKNS spectral problem and its time evolution equation are first generalized by embedding a finite number of differentiable and time-dependent functions. Starting from the generalized AKNS spectral problem and its generalized time evolution equation, a generalized AKNS hierarchy with variable coefficients is then derived. Furthermore, based on a systematic analysis on the time dependence of related scattering data of the generalized AKNS spectral problem, exact solutions of the generalized AKNS hierarchy are formulated through the inverse scattering transform method. In the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. It is graphically shown that the dynamical evolutions of such soliton solutions are influenced by not only the time-dependent coefficients but also the related scattering data in the process of propagations.     

Exponential rational function method for space–time fractional differential equations

In this paper, exponential rational function method is applied to obtain analytical solutions of the space–time fractional Fokas equation, the space–time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space–time fractional coupled Burgers’ equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.     

FUNCTIONAL REPRESENTATION OF THE NEGATIVE AKNS HIERARCHY

This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local systems of higher order, derive the generating function of the conservation laws and the N-dark-soliton solutions for the extended AKNS hierarchy. As an additional result we obtain the functional representation of the Landau–Lifshitz hierarchy.     

New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

In this paper, the ansatz method and the functional variable method are employed to find new analytic solutions for the space–time nonlinear fractional wave equation, the space–time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation and the space–time fractional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, some exact solutions are obtained in terms of hyperbolic and periodic functions. It is shown that the proposed methods provide a more powerful mathematical tool for constructing exact solutions for many other nonlinear fractional differential equations occurring in nonlinear physical phenomena. We have also presented the numerical simulations for these equations by means of three dimensional plots.     

Computational soliton solutions to $$$$-dimensional generalised Kadomtsev–Petviashvili and $$(2+1)$$(2+1)-dimensional Gardner–Kadomtsev–Petviashvili models and their applications

In this paper, we present a formalism to generate a family of interior solutions to the Einstein–Maxwell system of equations for a spherically symmetric relativistic charged fluid sphere matched to the exterior Reissner–Nordström space–time. By reducing the Einstein–Maxwell system to a recurrence relation with variable rational coefficients, we show that it is possible to obtain closed-form solutions for a specific range of model parameters. A large class of solutions obtained previously are shown to be contained in our general class of solutions. We also analyse the physical viability of our new class of solutions.     

Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV–Zakharov–Kuznetsov equation

In this paper, the first integral method and the functional variable method are used to establish exact traveling wave solutions of the space–time fractional Schrödinger–Hirota equation and the space–time fractional modified KDV–Zakharov–Kuznetsov equation in the sense of conformable fractional derivative. The results obtained confirm that proposed methods are efficient techniques for analytic treatment of a wide variety of the space–time fractional partial differential equations.     

Dynamical analysis of diversity lump solutions to the (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure equation

The lump solution is one of the exact solutions of the nonlinear evolution equation. In this paper, we study the lump solution and lump-type solutions of (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure (AKNS) equation by the Hirota bilinear method and test function method. With the help of Maple, we draw three-dimensional plots of the lump solution and lump-type solutions, and by observing the plots, we analyze the dynamic behavior of the (2+1)-dimensional dissipative AKNS equation. We find that the interaction solutions come in a variety of interesting forms.     

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.     

Wide-Angle Parabolic Approximation for Modeling High-Intensity Fields from Strongly Focused Ultrasound Transducers

A novel numerical algorithm based on the wide-angle parabolic approximation is developed for modeling linear and nonlinear fields generated by axially symmetric ultrasound transducers. An example of a strongly focused single-element transducer is used to compare the results of ultrasound field simulations based on the Westervelt equation, Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with differently modified boundary condition, and nonlinear wide-angle parabolic equation. It is demonstrated that having a computational speed comparable to modeling the KZK equation, the use of wide-angle parabolic approximation makes it possible to obtain solutions for highly focused ultrasound beams that are closer in accuracy to solutions based on the Westervelt equation.     

Dynamic stiffness matrix of thin-walled composite I-beam with symmetric and arbitrary laminations

For the spatially coupled free vibration analysis of thin-walled composite I-beam with symmetric and arbitrary laminations, the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented. For this, a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force–displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters. Then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force–displacement relationships. In addition, the finite-element (FE) procedure based on Hermitian interpolation polynomials is developed. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with analytical solutions, the results from available references and the FE analysis using the thin-walled Hermitian beam elements. Particular emphasis is given in showing the phenomenon of vibrational mode change, the effects of increase of the modulus and the bending–twisting coupling stiffness for beams with various boundary conditions.     

Painlevé integrability and multi-dromion solutions of the 2+1 dimensional AKNS system

The Painlevé integrability of the 2+1 dimensional AKNS system is proved. Using the standard truncated Painlevé expansion which corresponds to a special B?cklund transformation, some special types of the localized excitations like the solitoff solutions, multi-dromion solutions and multi-ring soliton solutions are obtained. Received 31 January 2001 and Received in final form 15 May 2001     

New application to Riccati equation

<正>To seek new infinite sequence of exact solutions to nonlinear evolution equations,this paper gives the formula of nonlinear superposition of the solutions and Backlund transformation of Riccati equation.Based on the tanhfunction expansion method and homogenous balance method,new infinite sequence of exact solutions to Zakharov-Kuznetsov equation,Karamoto-Sivashinsky equation and the set of(2+l)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica.The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.     

Numerical simulation of parametric sound generation and its application to length-limited sound beam

In this study we propose a simulation model for predicting the nonlinear sound propagation of ultrasound beams over a distance of a few hundred wavelengths, and we estimate the beam profile of a parametric array. Using the finite-difference time-domain method based on the Yee algorithm with operator splitting, axisymmetric nonlinear propagation was simulated on the basis of equations for a compressible viscous fluid. The simulation of harmonic generation agreed with the solutions of the Khokhlov–Zabolotskaya–Kuznetsov equation around the sound axis except near the sound source. As an application of the model, we estimated the profiles of length-limited parametric sound beams, which are generated by a pair of parametric sound sources with controlled amplitudes and phases. The simulation indicated a sound beam with a narrow truncated array length and a width of about one-quarter to half that of regular a parametric beam. This result confirms that the control of sound source conditions changes the shape of a parametric beam and can be used to form a torch like low-frequency sound beam.     

Continuous Correspondence of Conservation Laws of the Semi-discrete AKNS System

In this paper we investigate the semi-discrete Ablowitz–Kaup–Newell–Segur (sdAKNS) hierarchy, and specifically their Lax pairs and infinitely many conservation laws, as well as the corresponding continuum limits. The infinitely many conserved densities derived from the Ablowitz-Ladik spectral problem are trivial, in the sense that all of them are shown to reduce to the first conserved density of the AKNS hierarchy in the continuum limit. We derive new and nontrivial infinitely many conservation laws for the sdAKNS hierarchy, and also the explicit combinatorial relations between the known conservation laws and our new ones. By performing a uniform continuum limit, the new conservation laws of the sdAKNS system are then matched with their counterparts of the continuous AKNS system.     

Nonlinear Schrödinger equations and simple Lie algebras

We associate a system of integrable, generalised nonlinear Schrödinger (NLS) equations with each Hermitian symmetric space. These NLS equations are considered as reductions of more general systems, this time associated with a reductive homogeneous space. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is investigated using “r-matrix” techniques and shown to be “canonical” for all these equations. Throughout the reduction procedure this Hamiltonian structure does not degenerate. Each of the above systems of equations is gauge equivalent to a generalised ferromagnet. Reductions of the latter are discussed in terms of the corresponding NLS type equations.     

On Heisenberg models with values in Hermitian symmetric spaces

Heisenberg models with values in Hermitian symmetric spaces are introduced. A matrix form of the corresponding nonlinear Schrödinger equations is also presented. It is shown that a hidden symmetry of the generalized Heisenberg models generates a one-parameter family of solutions.     

Random Matrices with Equispaced External Source

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 × 1 vector-valued Riemann–Hilbert problems, and to perform an asymptotic analysis of the Riemann–Hilbert problems.