Nonzero Solutions Research Articles

TECHNIQUES TO SOLVE UNIFORM THIRD DEGREE EQUATION HAVING FOUR VARIABLES

Each article should be accompanied by an abstract of up to 250 words in English, which should mention all the principal facts and conclusions set forth in the paper. A similar abstract in Hindi language may be added, but this is not required. The homogeneous cubic equation with four unknowns given by is studied for obtaining its non-zero distinct integer solutions. Substitution technique and factorization method are utilized to determine the same.

Rogue waves and nonzero background solutions for the Gross–Pitaevskii equation with a parabolic potential

In this paper an integrable Gross–Pitaevskii equation with a parabolic potential and a gain term is investigated. Its solutions with a nonzero background are derived. These solutions are constructed by using biliearization reduction approach and connections between the nonlinear Schrödinger equation and the Gross–Pitaevskii equation. The solutions are presented in double-Wronskian form and are classified in terms of canonical forms of a certain matrix. Various breathers and rogue waves are analyzed and illustrated.

On the lifespan of nonzero background solutions to a class of focusing nonlinear Schrödinger equations

The global solvability in time and the potential for blow-up of solutions to non-integrable focusing nonlinear Schrödinger equations with nonzero boundary conditions at infinity present challenges that are less explored and understood compared to the case of zero boundary conditions. In this work, we address these questions by establishing estimates on the lifespan of solutions to non-integrable equations involving a general class of nonlinearities. These estimates depend on the size of the initial data, the growth of the nonlinearity, and relevant quantities associated with the amplitude of the background. The estimates provide quantified upper bounds for the minimum guaranteed lifespan of solutions. Qualitatively, for small initial data and background, these upper bounds suggest long survival times consistent with global existence of solutions. On the other hand, for larger initial data and background, the estimates indicate the potential for the intriguing phenomenon of instantaneous collapse in finite time. These qualitative theoretical results are illustrated via numerical simulations. Furthermore, importantly, the numerical findings motivate the proof of improved theoretical upper bounds that provide excellent quantitative agreement with the order of the numerically identified lifespan of solutions.

Binary Third degree Diophantine Equation 5 (x-y)3 = 8xy

This article emphasizes on finding non-zero different integer solutions to binary third degree Diophantine equation 5 (x-y)3 = 8xy . Two different sets of solutions in integers are presented. Some fascinating relations from the solutions are obtained. The method to get second order Ramanujan numbers is illustrated.

On Solving Cubic Equation x^3+y^3 =7(z-w)^2 (z+w)

The cubic Quadratic Equation is analyzed for its non-zero distinct integer solutions. Five different patterns of non-zero distinct integer solutions to the equation under consideration are obtained. A few applications of Diophantine equations are also presented.

A Portrayal of Integer Solutions to Transcendental Equation with Six Unknowns 𝟑√ 𝐗 + 𝐘 + √ 𝐱𝐲 = 𝑧𝟐 + 𝑤𝟐

Objectives: This research article focuses on finding non-zero integer solutions to the Transcendental equation with six unknowns represented by . Methods: There is no universal method for finding integer solutions to Diophantine equations involving surds. Various substitutions strategies are employed to obtain non-zero distinct integer solutions to the surd equation under consideration. Findings: Five different choices of substitutions are utilized to determine many non-zero distinct integer solutions to the transcendental equation with six unknowns in title. Novelty: In this analysis, the transcendental equation with six unknowns given by has been reduced to either Space Pythagorean equation or non-homogeneous ternary quadratic equation through suitable transformations and for which integer solutions can be found elegantly. Keywords: Transcendental Equation With Six Unknowns, Surd Equation, Integer Solutions, Space Pythagorean Equation, Ternary Quadratic Equation

On a family of nonzero solutions to the heat equation ut = △u on ℝn+1 which vanish on an arbitrary n-dimensional hyperplane P ⊂ ℝn+1

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data (i.e. vanishing on the hyperplane [Formula: see text]) can be easily obtained as a consequence of the examples on [Formula: see text] Finally, given an arbitrary hyperplane [Formula: see text], we can construct a family of nonzero solutions which vanish on [Formula: see text]

SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat {\cal{G}} $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $\hat {sl}(2,1)$ super-algebra.

On infinite spectra of oscillation exponents of third order linear differential equations

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

Observations On Ternary Quadratic Diophantine Equation

Abstract: The Ternary Quadratic Diophantine Equation is given by 2 2 2 8(x  y ) 15xy  7x  7 y  49  40z . Number of non-zero distinct integer solutions. Some interesting relations among the solutions and the way of factorization, along with the illusion of non-zero distinct integer solutions to the above equations are obtained.

Observations On Ternary Quadratic Diophantine Equation

Abstract: The Ternary Quadratic Diophantine Equation is given by 2 2 2 8(x  y ) 15xy  7x  7 y  49  40z . Number of non-zero distinct integer solutions. Some interesting relations among the solutions and the way of factorization, along with the illusion of non-zero distinct integer solutions to the above equations are obtained

Tackling Ternary Cubic Diophantine Equation

Abstract: A few interesting characteristics among the solutions and the patterns of non-zero integral solutions to the nonhomogeneous cubic equation with three unknowns represented by the Diophantine equation 2 2 3 4(   )  7( )  64 are examined.

Intrinsic Solution on Quadratic Diophantine Equation

Abstract: The non-zero unique integer solutions of the Quadratic Diophantine Equation with three unknowns

Intrinsic Solutions to the Biquadratic Diophantine Equations

Abstract: The Biquadratic Diophantine equation with five unknowns has non-zero unique intrinsic integral solutions obtained. The recurrence relation between the solutions is also obtained.

Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields

We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false.

Remarks on global Hölder regularity for a class of Monge-Ampère type equations

In this paper, we revisit the global Hölder regularity of nonzero convex solutions to the Dirichlet problem for a class of Monge-Ampère type equations. Based on the comparison principle, by choosing delicate auxiliary functions to construct the subsolutions, we extend the boundary regularity obtained in [Li-Li, Sci. China Math. 65 (3) (2022)] to more general equations, in which the parameters α,β,γ in the boundness condition are less restrictive. Also, the non-decreasing condition there can be replaced with a homogeneous type condition.

Elliptic equations with critical and supercritical growth at the boundary

In this study, we consider two classes of elliptic problems with nonlinear boundary conditions of concave–convex type. In the first problem, we obtain two nonzero and nonnegative solutions when the nonlinear term exhibits critical growth. In the second problem, we obtain infinitely many solutions (with no prescribed sign) by assuming that the nonlinearity is even and subcritical near the origin but with no growth condition at infinity.

Solving systems of homogeneous linear algebraic equations in pricing problems of an alternative financial model

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Theoretical and numerical investigation of a modified ABC fractional operator for the spread of polio under the effect of vaccination

<abstract> <p>The current manuscript investigates a model of the spread of polio under the condition of vaccination by using the novel modified Atangana-Baleanu-Caputo (mABC) fractional derivative. This problem has been studied for non-zero solutions under the modified operator. The series-type solution has been obtained through the application of a Laplace transformation, along with the decomposition technique and Adomian polynomial for the nonlinear terms. The qualitative analysis for the solution of the model has been tested by using fixed point theory. The stability of the solution is also crucial for a dynamical system; therefore, it was checked by using the T-Picard method. With the help of the approximate scheme, numerical simulations were conducted for the proposed model by using different fractional orders and transmission parameters. Based on the obtained positivity of the solutions and numerical stability, we have established the analysis of the mABC operator in the field of fractional calculus and other physical sciences.</p> </abstract>

Construction by the Tikhonov Method of a Nonzero Solution of the Homogeneous Cauchy Problem for one Equation with a Fractional Derivative

Construction by the Tikhonov Method of a Nonzero Solution of the Homogeneous Cauchy Problem for one Equation with a Fractional Derivative