A new extended Riccati equation rational expansion method and its application

In this paper, a new extended Riccati equation rational expansion method is suggested to constructing multiple exact solutions for nonlinear evolution equations. The validity and reliability of the method is tested by its application to the dispersive long wave system and the Broer–Kaup–Kupershmidt system. The method can be applied to other nonlinear evolution equations in mathematical physics.     

The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation

By using the Chen et al. ansatz [Chen Y, Wang Q, Lang Y. Naturforsch 2005;60a:127] and by modifying our extended Fan sub-equation method [Yomba E. Phys Lett A 2005;336:463]. We have obtained new and more general solutions including a series of non-travelling wave and coefficient function solutions namely: soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions for the (2+1)-dimensional Broer–Kaup–Kupershmidt equation. The most important achievement of this method lies on the fact that we have succeeded in one move to give all the solutions which can previously be obtained by application of at least four methods (the method using the Riccati equation, or the first kind elliptic equation, or the auxiliary ordinary equation, or the generalized Riccati equation as mapping equation).     

Consistent Riccati expansion for finding interaction solutions of (2+1)‐dimensional modified dispersive water‐wave system

A consistent Riccati expansion (CRE) is proposed to solve the (2+1)‐dimensional modified dispersive water‐wave (MDWW) system. It is proved that the MDWW system is CRE solvable. Furthermore, new exact interaction solutions, namely, soliton‐trigonometric waves, trigonometric waves‐soliton, soliton‐cosine periodic waves, and soliton‐cnoidal waves are explicitly derived.     

Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics

Conservation law plays a vital role in the study of nonlinear evolution equations, particularly with regard to integrability, linearization and constants of motion. In the present paper, it is shown that infinitely many conservation laws for certain nonlinear evolution equations are systematically constructed with symbolic computation in a simple way from the Riccati form of the Lax pair. Note that the Lax pairs investigated here are associated with different linear systems, including the generalized Kaup–Newell (KN) spectral problem, the generalized Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem, the generalized AKNS–KN spectral problem and a recently proposed integrable system. Therefore, the power and efficiency of this systematic method is well understood, and we expect it may be useful for other nonlinear evolution models, even higher-order and variable-coefficient ones.     

Variable-coefficient projective Riccati equation method and its application to a new (2 + 1)-dimensional simplified generalized Broer–Kaup system

In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Analytic solutions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system

Based on a Riccati equation and one of its new generalized solitary solutions constructed by the Exp‐function method, new analytic solutions with free parameters and arbitrary functions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system are obtained. These free parameters and arbitrary functions reveal that the (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system has rich spatial structures. As an illustrative example, two new spatial structures are shown by setting the arbitrary functions as different Jacobi elliptic functions. Compared with tanh‐function method and its extensions, the method proposed in this paper is more powerful and it can be applied to other nonlinear evolution equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Coupling Bäcklund trasnsformation of Riccati equation and division theorem method for traveling wave solutions of some class of NLPDEs

In this paper, a effective method for searching infinite sequence periodic and solitary wave solutions to nonlinear partial differential equations (NLPDEs) is proposed. A simple transformation technique and division theorem are used to reduce some class of NLPDEs to the Riccati equation, and then the infinite sequence periodic and solitary wave solutions of some class of NLPDEs are constructed by using Bäcklund transformation of Riccati equation and nonlinear superposition principle. As illustrative examples, we obtain the infinite sequence travelling-wave solutions of the three special equations, respectively.     

Residual symmetry,nth Bäcklund transformation,and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.     

On a 2+1‐Dimensional Whitham–Broer–Kaup System: A Resonant NLS Connection

It is established that the Whitham–Broer–Kaup shallow water system and the “resonant” nonlinear Schrödinger equation are equivalent. A symmetric integrable 2+1‐dimensional version of the Whitham–Broer–Kaup system is constructed which, in turn, is equivalent to a recently introduced resonant Davey–Stewartson I system incorporating a Madelung–Bohm type quantum potential. A bilinear representation is adopted and resonant solitonic interaction in this new 2+1‐dimensional Kaup–Broer system is exhibited.     

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.     

Exact traveling wave solutions of the Boussinesq–Burgers equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of the Boussinesq–Burgers equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many exact traveling wave solutions of the Boussinesq–Burgers equation are successfully obtained.     

Optimal State Feedback Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems

We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by state feedback. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a right factorization over H-infinity. A necessary and sufficient condition in terms of an (arbitrary) realization is that each state which can be reached in a finite time from the zero initial state has a finite cost. Another equivalent condition is that the control Riccati equation has a solution (in general unbounded and even non densely defined). The optimal state feedback input-output stabilization problem can then be solved explicitly in terms of the smallest solution of this control Riccati equation. We further show that after renorming the state space in terms of the solution of the control Riccati equation, the closed-loop system is not only input-output stable, but also strongly internally stable. Received: July 4, 2007. Revised: October 17, 2007.     

New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation

The repeated homogeneous balance method is used to construct new exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) 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 soliton-like and periodic-like solutions. This method is straightforward and concise, and it can be also applied to other nonlinear evolution equations.     

Nonlocal symmetry and soliton–cnoidal wave solutions of the Bogoyavlenskii coupled KdV system

The truncated Painlevé expansion is developed to construct Bäcklund transformations and nonlocal symmetries of the Bogoyavlenskii coupled KdV (BcKdV) system. The Schwarzian form of BcKdV system is found while the nonlocal symmetry is localized to offer the corresponding nonlocal group. Furthermore, the BcKdV system is verified to have a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, the soliton–cnoidal wave solutions are explicitly studied.     

Fractional quasi AKNS‐technique for nonlinear space–time fractional evolution equations

This paper aims to formulate the fractional quasi‐inverse scattering method. Also, we give a positive answer to the following question: can the Ablowitz‐Kaup‐Newell‐Segur (AKNS) method be applied to the space–time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi‐conservation laws for the considered fractional equations from the defined fractional quasi AKNS‐like system. The nonlinear fractional differential equations to be studied are the space–time fractional versions of the Kortweg‐de Vries equation, modified Kortweg‐de Vries equation, the sine‐Gordon equation, the sinh‐Gordon equation, the Liouville equation, the cosh‐Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.     

New exact solutions to the CDF equations

New exact soliton solutions to the Cologero–Degasperies–Fokas (CDF) equations in (1+1)-dimension and (2+1)-dimension by using the improved tanh method are investigated. First, the (1+1)-dimensional CDF equation is analyzed. By the improved tanh method, the corresponding nonlinear partial differential equation is reduced to the nonlinear ordinary differential equations and then the different types of exact solutions to the original equation are obtained based on the solutions of the Riccati equation. For the case of (2+1)-dimensional CDF equation the same computation procedure is carried out. It is presented that one could obtain new exact explicit solutions, which are traveling wave solutions, to (2+1)-dimensional CDF equation. Additionally, some graphical representations of the solitary and periodic solutions are presented.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.