# Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

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## Abstract

**:**

## Contents

**1. Introduction**

**2. Construction of Complex Solutions from Simple Solutions by Translation and Scale Transformations**

**3. Construction of Complex Solutions by Adding Terms or Combining Two Solutions**

**4. The Use of Complex-Valued Parameters for Constructing Exact Solutions**

**5. Using Solutions of Simpler Equations for Construct Solutions to Complex Equations**

**6. Brief Conclusions**

**References**

## 1. Introduction

#### 1.1. Preliminary Remarks

- simple exact solutions can serve as a basis for constructing more complex solutions of the equations under consideration,
- exact solutions to some equations can serve as the basis for constructing solutions to other more complex equations.

#### 1.2. Concept of ‘Exact Solution’ for Nonlinear PDEs

- (i)
- in terms of elementary functions, functions included in the equation (this is necessary when the equation contains arbitrary functions), and indefinite or/and definite integrals;
- (ii)
- through solutions of ordinary differential equations or systems of such equations.

## 2. Construction of Complex Solutions from Simple Solutions by Translation and Scale Transformations

#### 2.1. Some Definitions. Simplest Transformations

**Example**

**1.**

#### 2.2. Construction of Complex Solutions from Simpler Solutions. Examples

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Proposition**

**3.**

**Proof.**

**Example**

**8.**

**Example**

**9.**

#### 2.3. Generalization to Nonlinear Multidimensional Equations

**Example**

**10.**

#### 2.4. Generalization to Nonlinear Systems of Coupled Equations

**Example**

**11.**

**Example**

**12.**

## 3. Construction of Complex Solutions by Adding Terms or Combining Two Solutions

#### 3.1. Construction of Complex Solutions by Adding Terms to Simpler Solutions

**Example**

**13.**

**Remark**

**4.**

**Example**

**14.**

**Example**

**15.**

#### 3.2. Construction of Compound Solutions (Nonlinear Superposition of Solutions)

**Example**

**16.**

**Example**

**17.**

## 4. The Use of Complex-Valued Parameters for Constructing Exact Solutions

#### 4.1. Linear Partial Differential Equations

**Proposition**

**4.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Example**

**18.**

**Example**

**19.**

**Example**

**20.**

**Example**

**21.**

**Proposition**

**5.**

**Corollary**

**3.**

**Remark**

**5.**

#### 4.2. Nonlinear Partial Differential Equations

**Example**

**22.**

**Example**

**23.**

**Remark**

**6.**

**Proposition**

**6.**

**Example**

**24.**

## 5. Using Solutions of Simpler Equations for Construct Solutions to Complex Equations

**Preliminary remarks.**It is often possible to use solutions of simpler equations to construct exact solutions to complex differential equations. In this section, we will illustrate the reasoning in such cases for nonlinear PDEs (see Section 5.1), as well as for more complex nonlinear partial functional differential equations (see Section 5.2, Section 5.3 and Section 5.4).

#### 5.1. Nonlinear Partial Differential Equations

**Example**

**25.**

**Proposition**

**7.**

**Remark**

**7.**

**Example**

**26.**

#### 5.2. Partial Differential Equations with Delay

**Example**

**27.**

**Example**

**28.**

**Example**

**29.**

**Example**

**30.**

#### 5.3. Pantograph-Type Partial Differential Equations

**Example**

**31.**

**Example**

**32.**

**Remark**

**9.**

**Example**

**33.**

**Example**

**34.**

**Example**

**35.**

#### 5.4. Approach for Constructing Exact Solutions of Functional Partial Differential Equations

**The principle of analogy of solutions.**Structure of exact solutions to partial functional-differential equations of the form

**Example**

**36.**

**Example**

**37.**

**Remark**

**10.**

## 6. Brief Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Aksenov, A.V.; Polyanin, A.D. Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions. *Mathematics* **2021**, *9*, 345.
https://doi.org/10.3390/math9040345

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Aksenov AV, Polyanin AD. Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions. *Mathematics*. 2021; 9(4):345.
https://doi.org/10.3390/math9040345

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Aksenov, Alexander V., and Andrei D. Polyanin. 2021. "Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions" *Mathematics* 9, no. 4: 345.
https://doi.org/10.3390/math9040345