Explicit Exact Solutions Research Articles

Symmetry Reductions and Exact Solutions to the Kudryashov–Sinelshchikov Equation

Abstract In this article, based on the compatibility method, some nonclassical symmetries of Kudryashov–Sinelshchikov equation are derived. By solving the corresponding characteristic equations associated with symmetry equations, some new exact explicit solutions of Kudryashov–Sinelshchikov equation are obtained. For the exact explicit traveling wave solutions, the exact parametric representations are investigated by the integral bifurcation method.

Instability of equatorial edge waves in the background flow

In this paper, we first present an explicit exact solution to the edge wave problem in stratified geophysical flows with an underlying longshore current. Then we analyze the short-wavelength perturbation approach for barotropic incompressible fluids. Finally, we prove, by applying this method to geophysical equatorial edge waves in the background flow, that these waves are unstable when their steepness exceeds a specific threshold.

Momentum in strategic asset allocation

This paper explores a continuous-time intertemporal consumption and portfolio choice problem for an infinite horizon investor with recursive utility defined over consumption. The investor who tries to exploit momentum is assumed to have access to a risk-free asset and a risky asset whose return exhibits short run momentum. We derive an exact explicit solution and an approximate analytical solution to the dynamic asset allocation problem. We find that the optimal portfolio demand for stocks contains two components: the “momentum-adjusted” myopic demand and the intertemporal hedging demand. When the model is calibrated to Chinese stock market data, it implies that intertemporal hedging demand motives greatly decrease the portfolio demand for stocks by investors whose risk aversion coefficients exceed one when the latest levels of stock returns are non-negative or moderate negative. In addition, hedging motives increase the optimal portfolio when they are sufficiently negative. Also, we find that risk aversion is the main preference parameter in determining portfolio choice rather than the elasticity of intertemporal substitution.

The modified simple equation method and the multiple exp-function method for solving nonlinear fractional Sharma-Tasso-Olver equation

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructing the exact solutions and the solitary wave solutions for the nonlinear time fractional Sharma-Tasso- Olver equation. With help of Maple, we can get exact explicit 1-wave, 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions if we use the multiple exp-function method while we can get only exact explicit 1-wave solution including 1-soliton type solution if we use the modified simple equation method. Two cases with specific values of the involved parameters are plotted for each 2-wave and 3-wave solutions.

The large-time behaviour of coarsening of a particulate assemblage due to Ostwald ripening and coagulation

A theoretical description of the concluding stage of Ostwald ripening (coalescence) with allowance for coagulation of particles is considered. An implicit integral equation for the distribution function is derived for equal coagulation rates of the particles. This equation is analytically solved in the limit of large times. An exact explicit analytical solution for the volume density distribution function depends on the initial conditions and is in agreement with experimental data.

A new axis symmetric exact solution of the Maxwell’s equations in a time-variant nonlinear medium

A new explicit exact solution of the Maxwell’s equations for cylindrical wave in a time-variant nonlinear medium where is derived in this paper. We also prove that the solution seeking routine is effective for a large class of wave equations and exponential nonlinearity is necessary for additively separable solution. Physical interpretation of the additively separable solution is given too. Meanwhile, this method is valid if there exists a time-independent or a time-linear excited source.

Bäcklund transformation classification, integrability and exact solutions to the generalized Burgers’–KdV equation

This paper is concerned with the Bäcklund transformations (BTs) of the nonlinear evolution equations (NLEEs). Based on the homogeneous balance principle (HBP), the existence of the BT of the generalized Burgers’–KdV (B–KdV) equation is classified, then the BTs of the nonlinear equations are given. In general, the method can be used to construct BTs of the nonlinear evolution equations in polynomial form. Furthermore, the integrability and exact explicit solutions to the nonlinear equations are investigated.

Exact Nonlinear Internal Equatorial Waves in the f-plane

We present an explicit exact solution of the nonlinear governing equations for internal geophysical water waves propagating westward above the thermocline in the f-plane approximation near the equator. Moreover, the mass transport velocity induced by this internal equatorial wave is eastward and a westward current occurs in the transition zone between the great depth where the water is still and the thermocline.

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and anti-kink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schrödinger Equation

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

Exact traveling wave solutions and bifurcations of the generalized derivative nonlinear Schrödinger equation

In this paper, we consider a model of generalized derivative of nonlinear Schrodinger equation. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions under different parametric conditions. These results completely improve the study of traveling wave solutions for the mentioned model stated in Wang et al. (Commun Theor Phys 50:39–42, 2008).

Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges

Eigenstrains are created as a result of anelastic effects such as defects, temperature changes, bulk growth, etc., and strongly affect the overall response of solids. In this paper, we study the residual stress and deformation fields of an incompressible, isotropic, infinite wedge due to a circumferentially symmetric distribution of finite eigenstrains. In particular, we establish explicit exact solutions for the residual stresses and deformation of a neo-Hookean wedge containing a symmetric inclusion with finite radial and circumferential eigenstrains. In addition, we numerically solve for the residual stress field of a neo-Hookean wedge induced by a symmetric Mooney–Rivlin inhomogeneity with finite eigenstrains.

Gödel-type metrics in Einstein-Aether theory II: nonflat background in arbitrary dimensions

It was previously proved that the G\"{o}del-type metrics with flat three-dimensional background metric solve exactly the field equations of the Einstein-Aether theory in four dimensions. We generalize this result by showing that the stationary G\"{o}del-type metrics with nonflat background in $D$ dimensions solve exactly the field equations of the Einstein-Aether theory. The reduced field equations are the $(D-1)$-dimensional Euclidean Ricci-flat and the $(D-1)$-dimensional source-free Maxwell equations, and the parameters of the theory are left free except $c_{1}-c_{3}=1$. We give a method to produce exact solutions of the Einstein-Aether theory from the G\"{o}del-type metrics in $D$ dimensions. By using this method, we present explicit exact solutions to the theory by considering the particular cases: ($D-1$)-dimensional Euclidean flat, conformally flat, and Tangherlini backgrounds.

Painlevé analysis, $${\Phi }$$ Φ -integrable and exact solutions to the (3+1)-dimensional nonlinear evolution equations

This paper is concerned with the (3+1)-dimensional nonlinear evolution equation, which is an integro-differential equation actually. By constructing Backlund transformation of an auxiliary partial differential equation system through the Painleve test, the Backlund transformation of the original multi-dimensional nonlinear equation is presented under some condition, and the exact explicit solutions are provided accordingly. Moreover, based on the \(\Phi \)-expansion method, the definition of \(\Phi \)-integrable of the nonlinear differential system is proposed, and the exact explicit solutions are investigated.

New explicit solution for static shape control of smart laminated cantilever piezo-composite-hybrid plates/beams under thermo-electro-mechanical loads using piezoelectric actuators

In this paper, a new explicit exact analytical solution is proposed for obtaining static deformation and optimal shape control of smart laminated cantilever piezo composite hybrid plates and beams under thermo-electro-mechanical loads using piezoelectric actuators. The linear piezoelectricity and plates theories were adapted for the analysis. A novel double integral multivariable Fourier transformation method combined with discretised higher order partial differential unit step function equations were employed. The effect of various parameters including arbitrary loads such as non-uniform thermal stresses, electrical and mechanical loads, layup thickness, piezoelectric actuators size and placement, stacking sequence, and geometrical dimension were considered. The results were then compared with some published benchmark results and good agreement was observed. Unlike the earlier studies, the proposed method does not require the characteristic and trial deflection function to be predetermined. Both, the embedded and bounded actuators are considered. Until now, the shape control task of reducing mid-plane deformation at free end in smart laminated cantilever plates and beams was unsolvable and approximations were typically employed in numerical analysis Yu et al. (2009). This problem becomes more complicated for wider and longer plates but the method proposed herein successfully resolves this issue.

An exact explicit solution for one-dimensional, transient, nonlinear Richards’ equation for modeling infiltration with special hydraulic functions

Summary This work develops a simple exact and explicit solution of the one-dimensional transient and nonlinear Richards’ equation for soils in a special case of exponential water retention curve and power law hydraulic conductivity. The exact solution is obtained as traveling wave based on the approach proposed by Philip (1957, 1967) and adopted by Zlotnik et al. (2007). The obtained solution is novel, and it expresses explicitly the water content as function of the depth and time. It can be useful to model infiltration into semi-infinite soils with time-dependent boundary conditions and infiltration with constant boundary condition but space-dependent initial condition. A complete analytical inverse procedure based on the proposed analytical solution is presented which allows the estimation of hydraulic parameters. The proposed exact solution is also important for the verification of numerical schemes as well as for checking the implementation of time-dependent boundary conditions.

Explicit Exact Solution of Damage Probability for Multiple Weapons against a Unitary Target

Abstract We study the damage probability when M weapons are used against a unitary target. We use the Carleton damage function to model the distribution of damage probability caused by each weapon. The deviation of the impact point from the aimpoint is attributed to both the dependent error and independent errors. The dependent error is one random variable affecting M weapons the same way while independent errors are associated with individual weapons and are independent of each other. We consider the case where the dependent error is significant, non-negligible relative to independent errors. We first derive an explicit exact solution for the damage probability caused by M weapons for any M. Based on the exact solution, we find the optimal aimpoint distribution of M weapons to maximize the damage probability in several cases where the aimpoint distribution is constrained geometrically with a few free parameters, including uniform distributions around a circle or around an ellipse. Then, we perform unconstrained optimization to obtain the overall optimal aimpoint distribution and the overall maximum damage probability, which is carried out for different values of M, up to 20 weapons. Finally, we derive a phenomenological approximate expression for the damage probability vs. M, the number of weapons, for the parameters studied here.

Variable separation for time fractional advection-dispersion equation with initial and boundary conditions

In this paper, variable separation method combined with the properties of Mittag-Leffler function is used to solve a variable-coefficient time fractional advection-dispersion equation with initial and boundary conditions. As a result, a explicit exact solution is obtained. It is shown that the variable separation method can provide a useful mathematical tool for solving the time fractional heat transfer equations.

Lie symmetries and group classification of a class of time fractional evolution systems

In this paper, the Lie group method is developed to study a class of time fractional evolution systems. As an application, the complete group classification of a system of two component fractional evolution equations with an arbitrary function is performed. The admitted symmetries of all inequivalent systems belonging to the class under study are determined. What is more, all reduced fractional systems of ordinary differential equations and even explicit exact solutions are presented.