In this paper we present the tanh method to obtain exact solutions to coupled MkDV system. This method may be applied to a variety of coupled systems of nonlinear ordinary and partial differential equations.
We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic effective dynamic
s of solitons and their asymptotic stability. Results of numerical simulation are given. The obtained results allow us to formulate a new general conjecture on attractors of $G$ -invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohrs transitions between quantum stationary states, wave-particle duality and probabilistic interpretation.
Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Sever
This paper presents an observation that under reasonable conditions, many partial differential equations from mathematical physics possess three structural properties. One of them can be understand as a variant of the celebrated Onsager reciprocal re
lation in Modern Thermodynamics. It displays a direct relation of irreversible processes to the entropy change. We show that the properties imply various entropy dissipation conditions for hyperbolic relaxation problems. As an application of the observation, we propose an approximation method to solve relaxation problems. Moreover, the observation is interpreted physically and verified with eight (sets of) systems from different fields.
The method of characteristics has played a very important role in mathematical physics. Preciously, it was used to solve the initial value problem for partial differential equations of first order. In this paper, we propose a fractional method of cha
racteristics and use it to solve some fractional partial differential equations.
In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Greens function. In particular, we show that if the nonlinear term possesses a special multiplicativ
ity property, then its Greens function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies of specific non-linearities admitting this representation are derived. The nonlinear Greens function solution is numerically justified for the sinh-Gordon and Liouville equations. We also list two open problems leading to a more thorough characterizations of non-linearities admitting the obtained representation for the nonlinear Greens function.
Alvaro Sals Cesar Gomez (Universidad Nacional de Colombia
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(2008)
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"A practical approach to solve coupled systems of nonlinear partial differential equations"
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Alvaro Salas Humberto
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