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The compressible Euler equations in R N with Coriolis force is the fundamental mathematical model for the atmospheric dynamics. We use the vector and matrix technique to the N -dimensional Euler equations, to construct the following Cartesian vector form solutions u = b ( t ) + A ( t ) x . One advantage of this approach is that the Euler equations with Coriolis force can be solved both theoretically and algebraically, which can be accomplished by constructing appropriate matrices A ( t ) , and vector b ( t ) . We can obtain the new exact solutions using the required matrices A ( t ) and vector b ( t ) . As exact solutions for the compressible Euler equations are rare, the new exact solutions are valuable for the verification of the corresponding computational methods.

Reductions of the KP-Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev-Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP-Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios., 17 pages

The inverse mean curvature flow (IMCF) has been extensively studied not only as a type of geometric flows, but also for its applications to geometric inequalities. The focus is primarily on homothetic solitons for the IMCF in this paper, which are special solutions deformed only homothetically under the flow. We completely classify the profile curves of the higher dimensional rotationally and birotationally symmetric homothetic solitons using the phase-plane analysis. As a characterization of the round cylinder, we obtain that any helicoidal homothetic soliton of IMCF must be round cylinders. In addition, we prove that any surface foliated by circles to be the homothetic soliton of IMCF, which is a cyclic homothetic soliton, is either a surface of revolution or a piece of a round sphere.

In this paper, we have generalized the time-fractional telegraph equation involving operators with Mittag-Leffler kernel of variable order in Liouville-Caputo sense. The fractional variable-order equation was solved numerically via Crank-Nicholson scheme. We present the existence and uniqueness of the solution. Numerical simulations of the special solutions were done and new behaviors are obtained.

In this letter, we apply two different ansatzs for constructing the lump soliton and mixed lump strip solutions of (3+1)-dimensional soliton equation, which is associating with the Hirota bilinear form. These lump soliton solutions rationally localized in all directions in the space. The solutions of interactions between a lump and a stripe are shown by graphic illustration of some special solutions which would give us a better understanding on the evolution of solutions of waves. Keywords: Lump solutions, Mixed lump strip solutions, (3+1)-Dimensional soliton equation, Hirota bilinear form

The Szekeres system is studied with two methods for the determination of conservation laws. Specifically we apply the theory of group invariant transformations and the method of singularity analysis. We show that the Szekeres system admits a Lagrangian and the conservation laws that we find can be derived by the application of Noether's theorem. The stability for the special solutions of the Szekeres system is studied and it is related with the with the Left or Right Painlev\'e Series which describes the expansions., Comment: 6 pages, to be published in Phys. Lett. A

After publication of the paper, the author noticed that some mistakes occurred in formula (38) as explained in the main text.

Recently, Atangana and Baleanu introduced a novel derivative with fractional order with no singular and non-local kernel based on the Mittag–Leffler function. The new derivative is able to portray material heterogeneities and some structure or media with different scales. The non-locality of the new kernel allows better description of the memory within structure and media with different scales. To explicitly explain the reaction–diffusion within an alloy with different elements says, for instance, FeNi and NiFe, we make use of the newly defined fractional operator to enhance this model. We presented the special solution to the new model and some numerical simulations to appreciate the efficacy of the new derivative with fractional order.

In this paper, we establish exact special solutions of the phi-four equation. We use the reduction of order method and the He’s variational principle for this equation. So, we get general formulae for the compacton, solitary pattern, soliton, and periodic solutions.

In this work, a theoretical study of diffusion of neumatic liquid crystals was done using the concept of fractional order derivative. This version of fractional derivative is very easy to handle and obey to almost all the properties satisfied by the conventional Newtonian concept of derivative. The mathematical equation underpinning this physical phenomenon was solved analytically via the so-called homotopy decomposition method. In order to show the accuracy of this iteration method, we constructed a Hilbert space in which we proved its stability for the time-fractional Hunder-Saxton equation.

In connection with issues pertinent with humans’ health, it is highly significant to comprehend the complex dynamics of the related infectious disease since the non-Markovian effects play a vital role in its spread. This research analysis investigates an epidemiological model related with diarrhea transmission dynamics that occurred in Ghana during 2008–2018. The real data of 11 years have been obtained from Ministry of Health-Ghana and used to validate the proposed model. The epidemiological model is designed for the very first time with newly devised fractal fractional Caputo type operator having the fractional order α and the fractal dimension τ. The working parameters of the model are estimated and fitted with the real data using nonlinear least-squares fitting method and the basic reproduction number is found to be R 0 ≈ 2.09665 . Existence and uniqueness of the special solution of the proposed model are investigated. The disease free and the endemic equilibria are also computed. Finally, numerical simulations of the model are carried out via proposed numerical method and results in the graphical illustrations based upon effects of the estimated and fitted parameters show that the disease can either be substantially controlled or eradicated if the transmission rate of the epidemic is reduced and the rate of treatment is increased.

In this paper, we consider the discrete power function associated with the sixth Painlev\'e equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $\widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $\widetilde{W}(3A_1^{(1)})$ as a subgroup of $\widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{\rm VI}$, we show how to relate $\widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $\widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $\tau$ function., Comment: 18 pages, 3 figures

A tetrad field with six unknown functions is applied to the field equation of teleparallel equivalent of general relativity and analytic solution is derived. The coordinate $$\phi $$ to $$N(r,\theta ,\phi )$$ is generalized and the new anstaz to the field equation of $$f(T)$$ gravitational theories is applied. A special solution, with scalar torsion $$T={T^\alpha }_{\mu \nu } {S_\alpha }^{\mu \nu }=\hbox {constant}$$ , is derived with two constants of integration. The tetrad field of this solution is axially symmetric and its associated metric gives Kerr space-time.

We present the extension of the Einstein-Maxwell system called electrovac universes by introducing a cosmological constant $\Lambda$. In the absence of the $\Lambda$ term, the crucial equation in solving the Einstein-Maxwell system is the Laplace equation. The cosmological constant modifies this equation to become in a non-linear partial differential equation which takes the form $\Delta U=2\Lambda U^3$. We offer special solutions of this equation., Comment: 7 pages

Information theory is used in many branches of science and technology. For instance, to inform a set of human beings living in a particular region about the fatality of a disease, one makes use of existing information and then converts it into a mathematical equation for prediction. In this work, a model of the well-known river blindness disease is created via the Caputo and beta derivatives. A partial study of stability analysis was presented. The extended system describing the spread of this disease was solved via two analytical techniques: the Laplace perturbation and the homotopy decomposition methods. Summaries of the iteration methods used were provided to derive special solutions to the extended systems. Employing some theoretical parameters, we present some numerical simulations.

Quantum localization of classical mechanics within the BRST-BFV and BV (or field-antifield) quantization methods are studied. It is shown that a special choice of gauge fixing functions (or BRST-BFV charge) together with the unitary limit leads to Hamiltonian localization in the path integral of the BRST-BFV formalism. In turn, we find that a special choice of gauge fixing functions being proportional to extremals of an initial non-degenerate classical action together with a very special solution of the classical master equation result in Lagrangian localization in the partition function of the BV formalism., Comment: 11 pages, v2: Section 2 extended, a reference added, v3: typos corrected, v4: minor corrections, published version

The (G′/G)-expansion method and its simplified version are used to obtain generalized travelling wave solutions of five nonlinear evolution equations (NLEEs) of physical importance, viz. the (2+1)-dimensional Maccari system, the Pochhammer–Chree equation, the Newell–Whitehead equation, the Fitzhugh–Nagumo equation and the Burger–Fisher equation. A variety of special solutions like periodic, kink–antikink solitons, bell-type solitons etc. can easily be derived from the general results. Three-dimensional profile plots of some of the solutions are also drawn.

We study analytic properties of solutions to the q-Painleve VI equation (q-PVI), which was derived by Jimbo and Sakai as the compatibility condition for a connection preserving deformation (CPD) of a linear q-difference equation. We investigate local behaviours of solutions to q-PVI around a boundary point making use of the structure of the CPD. We also give a formula connecting the local behaviours of a solution around two boundary points. The results in this paper should be useful in future for studying more detailed global properties of solutions to q-PVI or exploring new special solutions with remarkable analytic properties.

An improved algorithm is devised for using the (G′/G)-expansion method to solve nonlinear differential-difference equations. With the aid of symbolic computation, we choose a discrete nonlinear Schrodinger equation to illustrate the validity and advantages of the improved algorithm. As a result, hyperbolic function solutions, trigonometric function solutions and rational solutions with parameters are obtained, from which some special solutions including the known solitary wave solution are derived by setting the parameters as appropriate values. It is shown that the improved algorithm is effective and can be used for many other nonlinear differential-difference equations in mathematical physics.

In this article, we give special solutions of second and fourth Painlev´e hierarchies derived by Gordoa, Joshi, and Pickering. We show that for certain choice of the parameters each n-th member of these hierarchies has a special solution in terms of an n-th order differential equation. Furthermore we derive a relation between these two hierarchies. In this article, we give special solutions of second and fourth Painlev´e hierarchies derived by Gordoa, Joshi, and Pickering. We show that for certain choice of the parameters each n-th member of these hierarchies has a special solution in terms of an n-th order differential equation. Furthermore we derive a relation between these two hierarchies.

With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV–MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations.

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order.

The q-difference Painleve II equation admits special solutions written in terms of determinant whose entries are the general solution of the q-Airy equation. An ultradiscrete limit of the special solutions is studied by the procedure of ultradiscretization with parity varialbes. Then we obtain new Airy-type solutions of the ultradiscrete Painleve II equation with parity variables, and the solutions have richer structure than the known solutions., Comment: 20 pages, 1 figure

A system of three equations which is called generalized Davey–Stewartson equations is considered. Some special solutions of these equations are investigated by both a tanh method and a variable separation approach.

Some special solutions of the Toda lattice model with a transversal degree of freedom are obtained. With the aid of Mathematica and Wu elimination method, more explicit solitary wave solutions, including compacton solutions, multiple compacton solutions, peakon solutions, as well as periodic solutions are found in this paper.

In this paper, we construct an analytic solution of the nonlinear modified Kortweg–de Vries–Burger equation. By applying a special solution of Cole–Hopf transformation to an ordinary differential equation, we propose a bounded travelling wave solution u(x, y, t) = U(ξ) where ξ = kx + ly − wt, to the two-dimensional nonlinear modified Kortweg–de Vries–Burger equation. We use Jacobian and rational methods to calculate the solitary wave solutions for the two-dimensional modified Kortweg–de Vries–Burger equation.

We consider families of one and a half degrees of freedom rapidly forced Hamiltonian systems which are perturbations of one degree of freedom Hamiltonians with a homoclinic connection. We derive the inner equation for this class of Hamiltonian system which is expressed as the Hamiltonian–Jacobi equation of a one a half degrees of freedom Hamiltonian. The inner equation depends on a parameter not necessarily small.We prove the existence of special solutions of the inner equation with a given behaviour at infinity. We also compute the asymptotic expression for the difference between these solutions. In some perturbative cases, this asymptotic expression is strongly related with the Melnikov function associated with our initial Hamiltonian.

In this paper, we study the nonlinear dispersive K(m, n) equations: ut + (um)x − (un)xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type, is developed. We prove that the existence of special solutions known as ‘double waves’ is equivalent to the diagonalizability of an arbitrary matrix of the two-parameter family Since the diagonalizability can be effectively verified by calculating the Haantjes tensor, this provides a simple necessary condition for integrability.

This paper presents a method to directly construct explicit exact solutions to nonlinear differential-difference equations. One applies this approach to solve Volterra lattice and Toda lattice and obtain their some special solutions which contain soliton solutions and periodic solutions.

In this paper, the BBM-like equations with fully nonlinear dispersion, B ( m , n ) equations: u t + ( u m ) x − ( u n ) xxt = 0 which exhibit solutions with compact support and with solitary patterns, are studied. The exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of the equations are found by a new method. The special cases, B (2, 2) and B (3, 3), are used to illustrate the concrete scheme of our approach presented by this paper in B ( m , n ) equations. The nonlinear equations B ( m , n ) are addressed for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of B ( m , n ) equations are established.

In this paper, we present a new method to obtain a new class of symmetries and invariant solution associated with the symmetries. The key point of our method is to find symmetry group of point transformations on a special solution. To demonstrate our method, we consider a spherically symmetric dynamics of an adiabatic gases, which describes a contracting and expanding localized mass of gas.

We present new non-Ricci-flat Kahler metrics with U(N) and O(N) isometries as target manifolds of superconformally invariant sigma models with an anomalous dimension. They are so-called Ricci solitons, special solutions to a Ricci-flow equation. These metrics explicitly contain the anomalous dimension and reduce to Ricci-flat Kahler metrics on the canonical line bundles over certain coset spaces in the limit of vanishing anomalous dimension., 9 pages, no figures

Based on Kodama–Gibbons approach we study hydrodynamic-type N reductions of the dispersionless Dym hierarchy. In particular, for N -truncated Lax operators, the hodograph coefficients associated with hodograph equations can be expressed in simple formulae. On the other hand, we also use Takasaki–Takebe approach to rederive the hodograph coefficients in twistor construction and discuss special solutions from deformations of twistor data. Our result elucidates the equivalence of the two approaches.

The multi-linear variable separation approach is applied to a differential-difference asymmetric Nizhnik-Novikov-Veselov equation. It is found that the solution formula ANNV equation is rightly the semi-discrete form of the continuous one which describes some types of special solutions for many (2+1)-dimensional continuous systems. Moreover, it is different but similar to that of a special differential-difference Toda system. Thus abundant semi-discrete localized coherent structures of the ANNV equation are easily constructed by appropriately selecting the arbitrary functions appearing in the final solution formula. A concrete method to construct multiple localized discrete excitations with and without completely elastic interaction properties are discussed. It is found that for some types of bounded solitoff modes, they can exchange solitoffs, move in new different directions, extend the lengths of solitoffs, etc. after finishing their interactions. - PACS numbers: 0230, 0220, 0540

In this paper, the Boussinesq-like equations with fully nonlinear dispersion, B(m,n) equations: utt+(um)xx−(un)xxxx=0 which exhibit solutions with solitary patterns, are studied. New exact solitary solutions of the equations are found. The two special cases, B(2,2) and B(3,3), are chosen to illustrate the concrete scheme of the decomposition method in B(m,n) equations. The nonlinear equations B(m,n) are addressed for two different cases, namely when m=n being odd and even integers. General formulas for the solutions of B(m,n) equations are established.

We classify all solutions of the restricted Toda chain from the ansatz that all moments are expressed in terms of polynomials of some variable. We show that each such solution corresponds to the Sheffer class orthogonal polynomials. It is shown that corresponding Hankel determinants are related to some well-known combinatorial numbers.

A bilinear variable separation approach is used to construct some special solutions for a differential-difference Toda equation. The semi-discrete form of the continuous formula which describes some types of special solutions for many (2 + 1)-dimensional continuous systems is found for a suitable quantity of the differential-difference Toda equation. Thus abundant semi-discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions.

We present Miura transformations for all discrete Painleve II equations known to date. We then use these Miuras to derive special solutions in terms of discrete Airy functions and to construct auto-Backlund transformations for the discrete Painleve equations. These transformations are then used to generate rational solutions. Some new forms of d-PII and d-P34 are obtained as well.

Transformation properties of perturbation expansions of vibrational quasi-periodic orbits upon the modular transformations of frequencies are studied, using simple but nontrivial examples: the Siegel complex quadratic map and special solutions of the real area-preserving quadratic map. It is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori, except for some special features of the real map, related to the atypical nature of 1/3 resonance in this case.

The (1+1)-dimensional Broer–Kaup system which describes the propagation of shallow water waves is extended to a generalized (2+1)-dimensional model with Painleve property. Some special exact solutions are obtained by using the truncated Painleve expansion. The generalized (2+1)-dimensional Broer–Kaup system also possesses quite rich localized excitations as other well-known (2+1)-dimensional integrable models. Especially, the elastic (pass through) interaction property without phase shift for two ring type solitons is graphically exhibited.

Using the compatibility condition for the Painleve equations, several new fourth-order ordinary differential equations (ODEs) that are analogies of the Painleve equations are found. The isomonodromic linear problems for these equations are given. Special solutions of the fourth-order ODEs found are discussed. The Painleve test is applied to investigate several fourth-order ODEs.

We find special solutions of the Rarita–Schwinger equation in spacetimes admitting shear free congruences of null geodesics.

The genuinely nonlinear dispersive K(m,n) equation, ut+(um)x+(un)xxx=0, which exhibits compactons: solitons with compact support, is investigated. New solitary-wave solutions with compact support are developed. The specific cases, K(2,2) and K(3,3), are used to illustrate the pertinent features of the proposed scheme. An entirely new general formula for the solution of the K(m,n) equation is established, and the existing general formula is modified as well.

In this paper, we study the genuinely nonlinear dispersive K(m,n) equation, ut−(um)x+(un)xxx=0, which exhibits solutions with solitary patterns. Exact solutions that create solitary patterns having cusps or infinite slopes are developed. The nonlinear equation K(m,n) is addressed for two different cases, namely when m=n=odd integer and when m=n=even integer. General formulas for the solutions of these cases of the K(m,n) equations are established.

We present fermionic extensions of the first two Painleve equations. In the case of P II , our starting point is the supersymmetric modified KdV equation to which we impose a similarity reduction. Various properties of the supersymmetric P II are studied, including special solutions and auto-Backlund transformations. For P I two different derivations are proposed, one based on a reduction of fermionic KdV and KP equations and another based on a direct search using singularity analysis.

The possible existence of a potentially chaotic soliton solution of an unperturbed classical oscillator is investigated. It is shown that in general the solution of the so-called “Helmholz oscillator” is a periodic solution. Special solutions are also found where a localized soliton solution or a decaying solution is obtained depending on the initial conditions.

We present a new family of the locus configurations which is not related to $\vee$-systems thus giving the answer to one of the questions raised in \cite{V1} about the relation between the generalised quantum Calogero-Moser systems and special solutions of the generalised WDVV equations. As a by-product we have new examples of the hyperbolic equations satisfying the Huygens' principle in the narrow Hadamard's sense. Another result is new multiparameter families of $\vee$-systems which gives new solutions of the generalised WDVV equation.

Matrix equations associated with the KP equation are considered and a kind of special solutions of them are found. The asymptotical behavior of the solutions along any direction is discussed and a sufficient condition for a solution vanishing along this direction is provided. Interaction between solutions of two different types can be represented by the solution.

We present the explicit formulae which allow to transform the general solution of the 6D Kaluza--Klein theory on a 3--torus into the special solution of the 6D bosonic string theory on a 3--torus as well as into the general solution of the 5D bosonic string theory on a 2--torus. We construct a new family of the extremal solutions of the 3D chiral equation for the SL(4,R)/SO(4) coset matrix and interpret it in terms of the component fields of these three duality related theories., Comment: 13 pages in LaTex