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For numerous fluids between elastic and viscous materials, the fractional derivative models have an advantage over the integer order models. On the basis of conformable fractional derivative and the respective useful properties, the bilinear form of time fractional Burgers equation and Boussinesq-Burgers equations are obtained using the generalized Bell polynomials and bilinear method. The kink soliton solution, anti-kink soliton solution, and the single-soliton solution for different fractional order are derived, respectively. The time fractional order system possesses property of time memory. Higher oscillation frequency appears as the time fractional order increasing. The fractional derivative increases the possibility of improving the control performance in complex systems with fluids between different elastic and viscous materials.

Based on the equations of the f-plane shallow water model, a new set of nonlinear models for the interaction of the vortices in the f-plane shallow water system are established by using perturbation expansion and multi-scale method. These models have different nonlinear characteristics. The asymptotic analytical solution(vortex solution) is obtained by using Newton iteration method and quasi-spectral method. In the simulation process, different from the previous studies, the initial values were constructed by using the asymptotic analytical solution for the first time to simulate the interaction process of the vortices. The result shows that the single vortex maintains a stable state for a long time under fixed parameters. The approximate critical distances of the binary vortices are different under different nonlinear effects. When the distance between binary vortices is less than, equal to, or greater than the critical distance, the binary vortices merge, rotate or separate. In addition, when the vortices interact, the critical distance of mKdV-type vortex is slightly larger than that of KdV-type vortex. This indicates that the nonlinear effect increases the energy loss rate during the interaction of the vortices and weakens the intensity of the interaction until the vortices are separated and do not affect each other.

Rossby waves, one of significant waves in the solitary wave, have important theoretical meaning in the atmosphere and ocean. However, the previous studies on Rossby waves commonly were carried out in the zonal area and could not be applied directly to the spherical earth. In order to overcome the problem, the research on ( $3+1$ )-dimensional Rossby waves in the paper is placed into the spherical area, and some new analytical solutions of ( $3+1$ )-dimensional Rossby waves are given through the classic Lie group method. Finally, the dissipation effect is analyzed in the sense of the above mentioned new analytical solutions. The new solutions on ( $3+1$ )-dimensional Rossby waves have important value for understanding the propagation of Rossby waves in the rotating earth with the influence of dissipation.

This paper proposes two stochastic SIQS epidemic models with periodic parameters and Markov switching. We first prove that the stochastic non-autonomous periodic system has a nontrivial positive periodic solution by using the Khasminskii’s theory. Then the sufficient conditions for extinction of the disease are obtained. Furthermore, we construct suitable stochastic Lyapunov functions with regime switching to prove the existence of ergodic stationary distribution of the stochastic SIQS epidemic model. At last, some rigorous numerical simulations are presented to illustrate our theoretical results.

For numerous fluids between elastic and viscous materials, the fractional magnetohydrodynamic models have an advantage over the integer-order models. We study the fractional magnetohydrodynamic equations with external force. On the basis of conformable fractional derivative and the respective useful properties, we study the well-posedness of the incompressible time fractional magnetohydrodynamic equations. In the qualitative study, the complicate nonlinear terms in time fractional magnetohydrodynamic equations are technical handled. The existence of the solution to the fractional incompressible magnetohydrodynamic equations is obtained. Moreover, the uniqueness and continuous dependence of the solution on the initial data are proved. In the quantitative study, using some mathematical transformation with practical physical significance, the exact solutions and the respective figures of the fractional magnetohydrodynamic equations are presented. The time fractional derivative system possesses property of time memory. The fractional derivative influences the control effect in complex systems with fluids between different elastic and viscous materials. The results are still valid if the time order is integer (integer-order magnetohydrodynamic equations case), or the magnetic field is zero (Navier–Stokes equations case).

In this paper, the exact solutions of a reduced extended $$(3+1)$$ -dimensional Jimbo–Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright–dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures.

We construct the exact nonlinear wave solutions of the Nonlinear Schrodinger equation on the period wave background instead of on a constant background. By using Darboux–Backlund transformation, soliton and breather solutions on two types of cnoidal wave backgrounds are given. The density evolutions of these solutions are given under different parameters to study their wave structures and dynamical properties.

These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.

This article studies the generalized Mittag&ndash, Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

The Kundu equation, which can be used to describe many phenomena in physics and mechanics, has crucial theoretical meaning and research value. In previous studies, the single Kundu equation has been investigated by the Riemann-Hilbert method, but few researchers have focused on the coupled Kundu equations. To our knowledge, many phenomena in nature can be only described by coupled equations, such as species competition and signal interactions. In this paper, we discuss N-soliton solutions of the coupled Kundu equations according to the Riemann-Hilbert method. Starting from the spectral problem, the coupled Kundu equations are generated, and the Riemann-Hilbert problem is presented. When the jump matrix of the Riemann-Hilbert problem is the identity matrix, the N-soliton solutions of the coupled Kundu equations can be expressed explicitly.

A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.

The (2+1)-dimensional incompressible and barotropic magnetohydrodynamic flow is studied combined with qualitative and quantitative analysis. The conservation of the energy of this system and the global existence of the solution ( U , b ) in L 2 ( [ 0 , + ∞ ) ; L 2 ( R 2 ) ) × L 2 ( [ 0 , + ∞ ) ; L 2 ( R 2 ) ) are proved utilizing priori estimate and Galerkin method. We present the general Lie symmetry analysis method to boundary value problem. The Lie symmetry analysis of magnetohydrodynamic equations with infinite boundary is proposed for finding infinitesimal generators, symmetry groups and various analytical solutions. We prove the nonlinear self-adjointness and compute the conservation law for the magnetohydrodynamic equations.

Much attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

A generalized (2+1)-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the (1+1)-dimensional KdV equation. TheN-soliton solutions of the (2+1)-dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficienteAij; wheneAij=0, the soliton fusion and fission will happen; wheneAij≠0, the pursuing collision will occur. Moreover, the Bäcklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method.

Frobenius integrable decompositions are introduced for nonlinear evolution equations with modified term. Two classes of polynomial equations with modified term are transformed into Frobenius integrable ordinary differential equations by two special Backlund transformations. The resulting solutions are illustrated to describe the solution phenomena shared with the mKdV equation, Sharma-Tasso-Olver equation, the modified Kawachara equation, the fifth-order KdV equation and its various forms.

A new 4-dimensional implicit vector-form loop algebra containing several arbitrary constants is presented, and the corresponding computing formula of constant γ in the Variation identity is obtained in this paper. As an application, a new Liouville integrable hierarchy is gotten which can be reduced to Dirac hierarchy, nonlinear Sch o ¨ inger equation (NLS), M-KdV equation and so on.

A Lie algebra is constructed and an isospectral problem with eight potentials is designed. From which, corresponding hierarchy of nonlinear evolution equation is derived and the integrable couplings of AKNS, D-AKNS, TD, Levi, Li, Ma and Fan hierarchies are deduced, and they are integrable in sense of Liouville. For each reduction case the relevant Hamiltonian structure is established by means of quadratic-form identity.

Starting from a new Lie algebra G 1 , the corresponding loop algebra G 1 ∼ is presented, from which a Liouville integrable hierarchy is given by using of variational identity. It follows that an expanding Lie algebra G 2 is obtained based on G 1 , furthermore, a related Lax integrable hierarchy is presented by making use of its related loop algebra G 2 ∼ . With the help of variational identity, it is not difficult to prove that the hierarchy has Hamiltonian structure and is Liouville integrable. It is also can be seen that the second hierarchy is the expanding model of the first one.

Two classes of nonlinear evolution equations are obtained, which possess Frobenius integrable decompositions. The resulting solitons are illustrated to describe the solution phenomena. (C) 2010 Elsevier B.V. All rights reserved.

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.

A Lie algebra containing four parameters is obtained, whose commutation operation is concise, and the corresponding computing formula of constant @c in the variational identity is presented in this paper. As application, a new Liouville integrable hierarchy which can be reduced to Dirac hierarchy is derived by designing a special isospectral problem. We call it generalized Dirac hierarchy.

Abundant new travelling wave solutions of the coupled Ito equation are obtained by the generalized Jacobi elliptic function method.

A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly, as its application, the multi-component NLS-MKdV equation hierarchy is obtained, then by use of trace identity the Hamiltonian structure of the above system is presented. Finally, the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.

A set of multi-component matrix Lie algebra is constructed, it follows that a type of new loop algebra is presented and multi-component integrable hierarchy is obtained. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system is worked out. As reduction of the hierarchy, some well-known hierarchy such as DNLS, KN, CLL hierarchy are established.

An algebraic system is constructed from which establishes two isospectral problems. By solving the zero curvature equations, two resulting integrable couplings of the Li hierarchy and Tu hierarchy are obtained, respectively. By making use of the quadratic-form identity, the Hamiltonian structures of the above integrable couplings are generated, which are Liouville integrable.

Integrable couplings of the Burgers hierarchy is obtained by the subalgebra of the loop algebra A ∼ 3 , then the Hamiltonian structure of the above system is worked out by the quadratic-form identity, finally the travelling wave solutions of the above system are presented by use of the tanh-function method.

Under the frame of the (2 + 1)-dimensional zero curvature equation and Tu model, (2 + 1)-dimensional Dirac hierarchy is obtained. Again by use of the expanding loop algebra the integrable coupling system of the above hierarchy is given.

Abundant new travelling wave solutions of the (2 + 1)-dimensional Sine–Gordon equation are obtained by the generalized Jacobi elliptic function method. The solutions obtained include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions.

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the NLS–MKdV hierarchy, and integrable coupling of multi-component Levi hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

A Liouville integrable system is obtained by the new subalgebra of the loop algebra A ∼ 3 , then the Hamiltonian structure of the above system is given by the quadratic-form identity. As a reduction case, Glachette–Johnson (GJ) hierarchy is presented.

Integrable couplings of the multi-component Dirac hierarchy is obtained by use of the vector loop algebra G ∼ M , then the Hamiltonian structure of the above system is given by the quadratic-form identity.

Integrable couplings of the Tu hierarchy is obtained by use of the new subalgebra of the loop algebra A ∼ 3 , then the Hamiltonian structure of the above system is given by the quadratic-form identity.

A set of multi-component matrix Lie algebra is constructed, it follows a type of new loop algebra that is presented. A (2 + 1)-dimensional multi-component AKNS integrable hierarchy is obtained by using a (2 + 1)-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system is worked out.

Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced.

Two high-dimensional Lie algebras are presented for which four (1+1)-dimensional expanding integrable couplings of the D-AKNS hierarchy are obtained by using the Tu scheme; one of them is a united integrable coupling model of the D-AKNS hierarchy and the AKNS hierarchy. Then (2+1)-dimensional DS hierarchy is derived by using the TAH scheme; in particular, the integrable couplings of the DS hierarchy are obtained.

We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1)-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalizedDp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value ofpis 5.

From rotational potential vorticity-conserved equation with topography effect and dissipation effect, with the help of the multiple-scale method, a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluids. By analyzing the equation, some conservation laws associated with Rossby solitary waves are derived. Finally, by seeking the numerical solutions of the equation with the pseudospectral method, by virtue of waterfall plots, the effect of detuning parameter and dissipation on Rossby solitary waves generated by topography are discussed, and the equation is compared with KdV equation and BO equation. The results show that the detuning parameterαplays an important role for the evolution features of solitary waves generated by topography, especially in the resonant case; a large amplitude nonstationary disturbance is generated in the forcing region. This condition may explain the blocking phenomenon which exists in the atmosphere and ocean and generated by topographic forcing.

Based on a well-known Lie algebra, the multicomponent Guo hierarchy with self-consistent sources is proposed. With the help of a set of non-semisimple Lie algebra, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources are obtained. It enriches the content of the integrable couplings of hierarchies with self-consistent sources. Finally, the Hamiltonian structures are worked out by employing the variational identity.