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The analytic multi-soliton solutions for nonlinear Schrodinger (NLS) equations are complex to obtain. Based on those solutions, interactions among multiple solitons show more abundant characteristics than two soliton interactions. With the Hirota method, bilinear forms and analytic soliton solutions of the coupled NLS equation are derived, and the influences of the dispersion parameter β(x) and constant parameters p1, p2 and p3 on soliton interactions are discussed in detail. The novel triple-S structures are presented via choosing suitable values. The phase, intensity and incidence angles of dark solitons are controlled with appropriate constant parameters. Besides, bound states of dark solitons are observed with different periods. In addition, the peculiar triple-triangle structures are presented when one sets β(x) as the hyperbolic tangent function. Results in this paper are useful for the generation and interaction of optical solitons in nonlinear optics and ultrafast optics.

In this article, we use the Hirota bilinear method. With the help of the symbolic calculation and applying the used method, we solve the (2+1)-dimensional Breaking Soliton (BS) equation. We obtain some interaction between lump soliton and solitary wave, the interaction between lump soliton and periodic wave, breather-type periodic soliton, periodic kink-wave, kink-soliton wave, and solitary wave solutions. All solutions have been verified back into its corresponding equation with the aid of Maple package program. The graphical representation of the solution is given by Maple and physically interpreted. The obtained results are useful in gaining the understanding of the various nonlinear scenarios in fluid dynamics and also the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering sciences.

In this paper, a fifth-order variable-coefficient nonlinear Schrodinger equation in an inhomogeneous optical fiber is investigated. Bilinear forms, which are different from those previously reported, are obtained under certain vraible-coefficient constraints. Modulational instability is shown to be related to the group velocity dispersion, Kerr nonlinearity and fifth-order dispersion. Dark soliton solutions are presented and discussed: Soliton velocity is related to the Kerr nonlinearity and fifth-order dispersion, while soliton amplitude is independent of them. Interactions between the dark two solitons are elastic, possibly the overtaking or head-on interactions. Soliton stability is also discussed via the numerical simulation, and the latter is verified through the independence verification.

The traveling wave reduction of the Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found. The Lax pair is used to obtain the first integrals of the traveling wave reduction for the Korteweg-de Vries hierarchy. Exact formulas for the first integrals of the hierarchy are given. The first three members of the hierarchy are considered in more detail. These first integrals are the examples of the integrable nonlinear differential equations with the Painleve property. Exact solutions in the form of the soliton for the traveling wave reduction of the Korteweg-de Vries hierarchy and its first integrals are presented.

In this paper, a reduced implicit sixth-order compact finite difference (CFD6) scheme which combines proper orthogonal decomposition (POD) technique and high-order compact finite difference scheme is presented for numerical solution of the Korteweg-de Vries (KdV) equation. High-order compact finite difference scheme is applied to attain high accuracy for KdV equation and the POD technique is used to improve the computational efficiency of the high-order compact finite difference scheme. This method is validated by considering the simulation of five examples, and the numerical results demonstrate that the reduced sixth-order compact finite difference (R-CFD6) scheme can largely improve the computational efficiency without a significant loss in accuracy for solving KdV equation.

One generalized Burgers hierarchy is derived by applying the Cole-Hopf transforation, whose dark-equation hierarchy is also generated by the dark-equation method, from which a generalized Burgers equation and a generalized Kupershmidt equation, respectively, are followed to obtain. Through Lie-group analysis method we produce similarity reductions, exact solutions of the generalized Burgers and the Kupershmidt equations. Specially, we investigate the similarity reductions of the fractional Kupershmidt equation and its exact solutions. In addition, we obtain the conservation laws of the Kupershmidt equation and its adjoint equation. Finally, we give rise to symmetries, primary branch solutions as well various recursion operators of degenerated equations from the Kupershmidt equation.

In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship.

We present a new spectral problem, a generalization of the D-Kaup–Newell spectral problem, associated with the Lie algebra sl( 2 , R ). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy and shows its Liouville integrability. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high-order symmetries.

Under investigation in this paper are the coupled Hirota (CH) equations, which describe the collision of two waves in the deep ocean and the propagation of the ultrashort optical pulses in a birefringent fiber. Based on the bilinear method, multi-soliton solutions for the CH equations are given. From the perspective of analysis, the interaction dynamics of solitons are obtained. The head-on and overtaking interactions of two/three solitons are analyzed. The elastic and inelastic interactions of two/three solitons are presented. The soliton velocity can be controlled by adjusting the physical parameters ϕ j , ( j = 1 , 2 , … N ) and ϵ . The energy exchange occurs between the variables u and v before and after the collision. The local interference of two/three solitons is observed. The closer to the center of the interaction, the more significant the interference. The real and imaginary parts of the parameters ϕ j , ( j = 1 , 2 , … N ) have the effects on the numbers of peaks and holes in the collisions. The structure of four eyes and one peak is observed during the two-soliton interaction. The three-soliton interactions are given with two peaks and local interference. It is hoped that the results of this study can provide some reference for the wave interaction in deep ocean, pulse propagation in optical fiber, financial option pricing, valuation of intangible assets in sports and so on.

The coupled derivative nonlinear Schrodinger III (cDNLSIII) equation describes pulse propagations in a two-mode nonlinear fiber. In this paper, we construct the N -fold generalized Darboux transformation (DT) with the limit approach, from which the general higher-order rogue wave solutions of the cDNLSIII equation are obtained. These solutions, divided into three categories, namely, rogue waves and rogue waves on multi-soliton and multi-breather backgrounds, have applications in nonlinear fiber communication. Furthermore, we present some interesting interactions between rogue waves and solitons or breathers and analyze the features of each type.

The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used simplest equation is f ξ 2 = n 2 ( f 2 − f ( 2 n + 2 ) / n ) . The developed methodology is illustrated on examples of classes of nonlinear partial differential equations that contain: (i) only monomials of odd grade with respect to participating derivatives; (ii) only monomials of even grade with respect to participating derivatives; (iii) monomials of odd and monomials of even grades with respect to participating derivatives. The obtained solitary wave solution for the case (i) contains as particular cases the solitary wave solutions of Korteweg–deVries equation and of a version of the modified Korteweg–deVries equation. One of the obtained solitary wave solutions for the case (ii) is a solitary wave solution of the classic version of the Boussinesq-type equation.

From the known Lax pair of the Gardner equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the Gardner equation with nonlocal symmetries by introducing suitable auxiliary dependent variables. Based on the prolonged system, the explicit analytic solutions related to the Jacobi elliptic functions are derived. Figures illustrate the interaction between the cnoidal waves and a kink wave.

In this paper, we investigate the integrable nonlocal pair-transition-coupled nonlinear Schrodinger equation with the help of the loop group method. By constructing the Darboux transformation associated with a variable separation technique, some new soliton solutions such as the doubly-periodic bright/dark/antidark soliton, X -shaped kink, rogue wave-like soliton, mixed type kink and double breather (hump) and different types of novel rational solutions are easily derived. The obtained results are different from the solutions of the local nonlinear equations and nonlocal nonlinear Schrodinger equation. With appropriate choices of free parameters, the wave structures have large changes. Furthermore, the dynamic characteristic of soliton solutions are discussed on the basics of figures.

In this work, we adopt the generalized Kudryashov method to produce exact wave solutions for some space-time fractional partial differential equations. The Jumaries modified Riemann-Liouville is considered. The proposed method is implemented to the fractional Sharma-Tasso-Olver equation and the fractional Bogoyavlenskii’s breaking soliton equations. Various solutions containing hyperbolic tangent function, trigonometric tangent function and exponential functions are obtained by this method. Simulations of some obtained solutions are given at the end of the paper.

Under investigation in this paper is the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like (CDGKS-like) equation. Based on bilinear neural network method, the generalized lump solution, classical lump solution and the novel analytical solution are constructed by giving some specific activation functions in the single hidden layer neural network model and the “3-2-2” neural network model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained with the help of Maple software. These results fill the blank of the CDGKS-like equation in the existing literature. Via various three-dimensional plots, curve plots, density plots and contour plots, dynamical characteristics of these waves are exhibited. The effective methods used in this paper is helpful to study the nonlinear evolution equations in plasmas, mathematical physics, electromagnetism and fluid dynamics.

We present a generalized study and characterization of the integrability properties of the derivative non-linear Schrodinger equation in 1 + 1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the singular manifold method. This procedure, together with the Darboux transformations, allow us to construct a wide class of rational soliton-like solutions. Clasical Lie symmetries have also been computed and similarity reductions have been analyzed and discussed.

Truncated Painleve expansions, invariant Painleve analysis, and generalized Hirota expansions are employed in combination to solve (‘partially reduce to quadrature’) the integrable KdV and KP equations, and a nonintegrable generalized microstructure (GMS) equation. Although the multisolitons of the KdV and KP equations are very well-known, the solutions obtained here for all the three NLPDEs are novel and non-trivial. The solutions obtained via invariant Painleve analysis are all complicated rational functions, with arguments which themselves are confluent hypergeometric (KdV) or trigonometric (GMS) functions of various distinct non-traveling (KdV) and traveling wave variables. In some cases, this is slightly reminiscent of doubly-periodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained by the use of recently-generalized Hirota-type expansions in the truncated Painleve expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the three NLPDEs considered here.

This paper considers the stochastic Korteweg–de Vries equation on a bounded domain. The existence of a weak martingale solution is discussed by the Galerkin’s approximation and compactness method. Based on this result, we establish the Unique Continuation Property for the stochastic Korteweg–de Vries equation.

In this paper, we study a nonintegrable discrete nonlinear Schrdinger (dNLS) equation with nonlinear hopping. By using the planar nonlinear dynamical map approach, we address the spatial properties of the nonintegrable dNLS equation. Through the constructions of exact period-1 and period-2 orbits of a planar nonlinear map which is a stationary version of the nonintegrable dNLS equation, we obtain the spatially periodic solutions of the nonintegrable dNLS equation. We also give the numerical simulations of the orbits of the planar nonlinear map and show how the nonlinear hopping terms affect those orbits. By using discrete Fourier transformation method, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation.

Combining the binary Bell polynomials and Hirota method, we obtained two kinds of equivalent bilinear equations for the Levi equation. Then, we got the double Wronskian solutions of the Levi equation by virtue of one of the bilinear equations. Furthermore, we constructed the bilinear Backlund transformation and the Lax pair. Finally, we also derived the Darboux transformation and the infinite conservation laws of the Levi equation.

Under investigation in this paper is a generalized variable-coefficient fifth-order Korteweg-de Vries equation, which describes the interaction between a water wave and a floating ice cover or the gravity-capillary waves. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear forms, N-soliton solutions, Backlund transformation and Lax pair are derived. Infinitely-many conservation laws are obtained based on the Bell-polynomial-typed Backlund transformation. Soliton fusion and fission, and influence of the variable coefficients from the equation are analyzed: Both variable coefficients c(t) and n(t) are in direct proportion to the soliton velocities but have no effect on the amplitudes, while another constant coefficient α can affect the types of the interactions, in the sense of the elastic or inelastic. Elastic-inelastic interactions among the three solitons are presented as well.

A new method to obtain second-order reductions for ordinary differential equations which are polynomial in the derivatives of the dependent variable is presented. The method is applied to obtain reductions and new solutions to several well-known equations of mathematical physics: a lubrication equation, a thin-film equation, the Zoomeron equation and a family of 5 th - order partial differential equations which includes the Caudrey-Dodd-Gibbon-Sawada-Kotera, Kaup-Kupershmidt, Ito and Lax equations. Some pieces of computer algebra code to derive the reductions are also included.

In this work, different kinds of solutions including breather-type and multi-wave solutions are obtained for the ( 2 + 1 ) -dimensional Gardner equation by using bilinear form, the extended homoclinic test approach and three-wave method. We obtained the coefficient conditions in solution ansatz for the existing of breather and multi-wave solutions. By selecting appropriate values of the parameter, three dimensional, contour and density plots of solutions are drawn in order to better understand the dynamic behaviors of considered physical phenomena.

A new integrable hierarchy related to a 4 × 4 matrix isospectral problem is proposed, in which the semi-discrete version of the coupled KdV system is the first member. Subsequently the N-fold Darboux transformation for the second member in the obtained hierarchy is constructed. As applications, the explicitly exact solutions to the equation in three cases of different seed solutions are discussed and their graphs are showed to analyze the corresponding dynamical properties. Meanwhile some numerical simulations are given to illustrate these properties.

An extended generalized Darboux transformation method is proposed to construct the hybrid rogue wave and breather solutions for a classical nonlinear Schrodinger equation. Three types of hybrid wave solutions are obtained: (i) the hybrid first-order rogue wave and breather; (ii) the hybrid second-order rogue wave and first-order breather; (iii) the hybrid first-order rogue wave and second-order breather. These solutions are novel and can be used to investigate the dynamical characteristic of the hybrid rogue waves and breathers. The control and interaction based on the parameters of the hybrid wave solution are graphically demonstrated. An exact link is established between the hybrid solutions and the rogue wave solutions via setting the parameter at special value.

In this paper, the solution of Benjamin-Bona-Mahony-Burgers (BBMB) equation is approximated by using collocation method. Stability of the method is discussed. Also the convergence of the method is proved. The method is used on some examples, and the numerical results have been obtained and compared with the exact solutions.

Under investigation in this paper is the integrability of a (3+1)-dimensional generalized KdV-like model equation, which can be reduced to several integrable equations. With help of Bell polynomials, an effective method is presented to succinctly derive the bilinear formalism of the equation, based on which, the soliton solutions and periodic wave solutions are also constructed by using Riemann theta function. Furthermore, the Backlund transformation, Lax pairs, and infinite conservation laws of the equation can easily be derived, respectively. Finally, the relationship between periodic wave solutions and soliton solutions are systematically established. It is straightforward to verify that these periodic waves tend to soliton solutions under a small amplitude limit.

The generalized modified Korteweg-de Vries equation of the fifth order with dissipation is considered. The Painleve test is applied for studying integrability of this equation. It is shown that the generalized modified Korteweg-de Vries equation of the fifth order does not pass the Painleve test in the general case but has the expansion of the solution in the Laurent series. As a consequence the equation can have some exact solutions at additional conditions on the parameters of the equation. We present the effective modification of methods for finding of solitary wave and elliptic solutions of nonlinear differential equations. Solitary wave and elliptic solutions of the generalized modified Korteweg-de Vries equation of the fifth order are found by means of expansion for solution in the Laurent series. These solutions can be used for description of nonlinear waves in the medium with dissipation, dispersion.

In this paper we consider the long time behavior of the three dimensional Benjamin-Bona-Mahony equations for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Benjamin-Bona-Mahony equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to V'. For the nonautonomous case, we only assume that f(x, t) is translation bounded instead of translation compact. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Benjamin-Bona-Mahony equations and also obtain the existence of the uniform attractor.

Korteweg-de Vries (KdV)-typed equations are seen in fluid dynamics, plasma physics and other fields. By means of the Bell-polynomial procedure, we take two higher-order members of the KdV hierarchy, the seventh- and ninth-order Lax's KdV equations in fluid dynamics, as the examples for studying the integrable properties of the higher-order equations. Different from lower-order equations, two new partial differential operators in the Bell-polynomial procedure are introduced to construct the "multi-dimensional" bilinear forms of such equations with several auxiliary independent variables. Through the procedure simplified via the algebraic operation of the polynomials, the Backlund transformations, Lax pairs and infinite conservation laws of such equations are deduced.

The modified Korteweg-de Vries equation with polynomial source is considered. Using the Painleve test we show that the generalized Korteweg-de Vries equation is not integrable by the inverse scattering transform. However there are some expansions of solution in the Laurent series and some exact solutions can exist. Some traveling wave solutions of the modified Korteweg-de Vries equation with polynomial source are found.

We are concerned with stable spectral collocation solutions to non-periodic Benjamin Bona Mahony (BBM), modified BBM and Benjamin Bona Mahony-Burgers (BBM-B) initial value problems on the real axis. The spectral collocation is based alternatively on the scaled Hermite and sinc functions. In order to march in time we use several one step and linear multistep finite difference schemes such that the method of lines (MoL) involved is stable in sense of Lax. The method based on Hermite functions ensures the correct behavior of the solutions at large spatial distances and in long time periods. In order to prove the stability we use the pseudospectra of the linearized spatial discretization operators. The extent at which the energy integral of BBM model is conserved over time is analyzed for Hermite collocation along with various finite difference schemes. This analysis has been fairly useful in optimizing the scaling parameter. The effectiveness of our approach has been confirmed by some numerical experiments.

In this paper, by using the simplest equation-Riccati equation and variable separation method, new exact solutions of (2 + 1)-dimensional Burgers equation are obtained. Based on the arbitrary function in the solutions, we also get various forms of the solutions, and some local soliton structure are discussed.

This work is devoted to the application of inverse spectral problem for integration of the periodic solutions of the discrete modified KdV equation with a self-consistent source. The effective method of solution of the inverse spectral problem for the discrete linear Ablowitz-Ladik equation is presented.

Based on the Lie superalgebras osp(2|2) and spo(2|2), we generate some new super soliton hierarchies. These hierarchies can reduce to the AKNS hierarchy and the super-AKNS hierarchy. And by using the supertrace identity on Lie superalgebras, we construct the super-Hamiltonian structures of these new super soliton hierarchies. Then, as the applications of spo(2|2) ≅ osp(2|2) and spo(2|2) ≅ sl(1|2), we obtain the relations of different hierarchies associated with different Lie superalgebras.

We investigate the existence and propagation properties of envelope solitons of an extended nonlinear Schr odinger equation with the time-modulated dispersion, quadratic-cubic nonlinearities and linear gain or loss, which govern the nonlinear wave propagation in a nonautonomous quasi-1D Bose-Einstein condensate system with contact repulsion and dipole-dipole attraction. A novel class of nonautonomous bright soliton solutions on continuous-wave background is identified for the first time. It is shown that these localized structures possess interesting features that differ from the usual bright solitons. A rich variety of evolution behaviors, which include snakelike and periodic oscillating bright soliton dynamics, is revealed. The constraints of the system parameters to form these nonlinear localized structures are also suggested.

In this paper, a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation, which can be reduced to the classical equation, is investigated based on the Hirota bilinear method. The lump and lump strip solutions for this equation are obtained with the help of symbolic computation. Those solutions are derived from polynomial solutions, and can be simply classified into some classes. Analysis for the obtained solutions are presented, and their dynamic properties are discussed. Results are helpful for the study of soliton interactions in nonlinear mathematical physics.

In this paper, we propose an integrable semi-discrete and full-discrete analogues of the modified coupled integrable dispersionless (mCID) equation that was derived recently as a two-component generalization of CID equation. The key of the integrable discretization is to find the relation between the mCID equation and the sine-Gordon equation. We complete the integrable discrete analogues based on the discrete sine-Gordon equation. Numerical simulation of one-soliton, two-soliton and breather solution are conducted based on the integrable semi-discrete scheme. We also present the Casorati determinant solution in parametric form of N-soliton solutions for the semi-discrete analogue of the mCID equation. We show that the continuum limit of the full-discrete mCID equation leads to the semi-discrete mCID equation.

This paper concerns the qualitative behavior and exact traveling wave solutions of the generalized double combined sinh–cosh-Gordon equation. This equation generalizes sinh-Gordon equation, Liouville equation, Dodd–Bullough–Mikhailov equation, Tzitzeica–Dodd-Bullough equation and Zhiber-Shabat equation as special cases. Thus, this paper presents a unified analysis to find the exact solutions of these known equations. Our results generalize many previous known results (Chen et al., 2009; Fan et al., 2011; Geng et al., 2007; He and Meng, 2017; He et al. 2014; Li and Li 2005; Seadawy et al. 2017; Tang and Huang 2007). We find different kinds of exact solutions such as bright soliton, dark soliton, kink wave, anti-kink wave solutions and periodic wave solutions. Moreover, the explicit expressions of the bounded exact traveling wave solutions are given. Finally, a conclusion ends this paper.

A bicentric quadrilateral is a tangential cyclic quadrilateral. In a tangential quadrilateral, the four sides are tangents to an inscribed circle, and in a cyclic quadrilateral the four vertices lie on a circumscribed circle. In this paper, we classify all planar central configurations of the 4-body problem, where the four bodies are at the vertices of a bicentric quadrilateral.

The same result of different ansatz based on extended tanh-function method for nonlinear models is firstly reported. As an example, we obtain eleven kinds of variable separation solutions of variable-coefficient Boiti-Leon-Pempinelli system based on the extended tanh-function method by considering three different ansatz, namely, positive power-form ansatz, positive and negative power-symmetric ansatz and radical sign combined ansatz. By careful analysis, we find that these seemly independent variable separation solutions are essentially same. From this perspective, different ansatz are not really effective to construct so-called "new" solutions. Therefore, we should check carefully solutions obtained by the same method with different ansatz, and avoid casually asserting these essentially same "new" solutions.

Based on the symmetric, nonnegative and normalized basis of the trigonometric polynomial space, the piecewise trigonometric Hermite interpolation methods are presented. The C n - 1 and Cn continuous piecewise trigonometric Hermite interpolants of degree n are constructed and the interpolation methods are local. The integral and the differential representations of the errors of the trigonometric Hermite interpolants are given. Several examples are supplied to support the practical value of the given interpolation methods.

Darboux transformation for a nonlinear integrable equation associated with 3 × 3 matrix spectral problems, which is constructed by Lenells, is derived with the aid of the gauge transformation between the corresponding Lax pairs. By using the Darboux transformation, new explicit solutions for the nonlinear integrable equation are given and their figures are plotted. Finally, infinitely many conservation laws of this equation are deduced.

An isospectral problem with a parameter ? is presented, for which a modified KdV (mKdV) hierarchy is easily derived from the Tu scheme. Then two different expanding integrable models are obtained, respectively. The main purpose for this focuses on deriving a (2+1)-dimensional mKdV hierarchy (called the mKP hierarchy) and the corresponding different (2+1)-dimensional expanding models (including the linear and nonlinear), whose Hamiltonian structures are obtained by employing an identity developed by us in the paper. As the reduced consequences, the linear and nonlinear (2+1)-dimensional expanding models of the mKP equation are generated.

In this article we explicitly construct Stackel separable systems in separation coordinates with the help of separation curve as introduced by Sklyanin. Further, we construct explicit transformation between separable and flat coordinates for flat Stackel systems. We also exploit the geometric structure of these systems in the obtained flat coordinates. These coordinates generalize the wellknown generalized elliptic and generalized parabolic coordinates introduced by Jacobi. (C) 2015 Elsevier Inc. All rights reserved. Funding Agencies|Royal Swedish Academy of Sciences [FOA 13Magn-088]

In this article, analytical solutions and different types of soliton envelopes: bright, dark and singular for the nonlinear model, namely, nanobioelectronics transmission lines have been constructed along with constrained conditions. The modified extended tanh-function method and exp-function method have been used to find analytical solutions, and while solitary wave ansatz is used to construct these soliton solutions. Additionally, the constraint conditions, for the existence of the soliton solutions are also listed.

General high-order rogue waves of the KP equation with self-consistent sources (KPESCSs) are derived via the Hirota bilinear method, which are given in terms of determinants whose matrix elements have plain algebraic expressions. By means of the regulation of free parameters, the presentation of fundamental rogue waves and the second-order rogue waves is demonstrated by the density and the three dimensional figures.

In this paper, two methods are employed to study the nonlinear wave solutions for two (3+1)-dimensional equations which can be reduced to the potential KdV equation.Firstly, using the simplified Hirota's method, we present generalized multiple soliton solutions and generalized multiple singular soliton solutions in which some differentiable arbitrary functions are involved. Secondly, by means of some special orbits of the traveling wave system and integrating approach, we obtain some other nonlinear wave solutions which also include differentiable arbitrary functions. Our work extends pioneer's results.

Under investigation in this paper is the canonical AB system, which describes the marginally unstable baroclinic wave packets in the geophysical fluids and ultra-short pulses in nonlinear optics. Through symbolic computation, conservation laws are derived. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Elastic interactions of two one-peak bright solitons, of two one-peak dark solitons and of two two-peak dark solitons; (2) Parallel propagations of two one-peak bright solitons, of two one-peak dark solitons and of two two-peak dark solitons; (3) Propagations of three types of bound solitons: periodic propagation of bound solitons taking on contrary trends, mutual attractions and repulsions of two bight bound solitons and of two dark bound solitons.

We apply the proper orthogonal decomposition (POD) to the nonlinear Schrodinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid-point rule. A priori error estimates are derived for the POD reduced dynamical system. Numerical results for one and two dimensional NLS equations, coupled NLS equation with soliton solutions show that the low-dimensional approximations obtained by POD reproduce very well the characteristic dynamics of the system, such as preservation of energy and the solutions. (C) 2015 Elsevier Inc. All rights reserved.